/:'>  s^f 


l/../\fiv£fi:^rr'i> 


. ' 6>"  ■ 
':■  >*  /»■ 


i 


^ --*!•■  V>- 


■ /'»V. 


WENTWORTH-SMITH  MATHEMATICAL  SERIES 


PLANE  TRIGONOMETRY 
AND  TABLES 


BY 

GEORGE  WENTWORTH 

AND 

DAVID  EUGENE  SMITH 


GINN  AND  COMPANY 


BOSTON  • NEW  YORK  • CHICAGO  • LONDON 
ATLANTA  • DALLAS  • COLUMBUS  • SAN  FRANCISCO 


COPYRIGHT,  1914,  BY  GEORGE  WENTWORTH 
AND  DAVID  EUGENE  SMITH 
ALL  RIGHTS  RESERVED 
PRINTED  IN  THE  UNITED  STATES  OF  AMERICA 
226.1 


SCbensum 


GINN  AND  COMPANY  • PRO- 
PRIETORS • BOSTON  • U.SA. 


PKEFACE 


In  preparing  a work  to  replace  the  Wentworth  Trigonometry, 
which  has  dominated  the  teaching  of  the  subject  in  America  for  a 
whole  generation,  some  words  of  explanation  are  necessary  as  to  the 
desirability  of  the  changes  that  have  been  made.  Although  the  great 
truths  of  mathematics  are  permanent,  educational  policy  changes  from 
generation  to  generation,  and  the  time  has  now  arrived  when  some 
rearrangement  of  matter  is  necessary  to  meet  the  legitimate  demands 
of  the  schools. 

The  principal  changes  from  the  general  plan  of  the  standard  texts 
in  use  in  America  relate  to  the  sequence  of  material  and  to  the 
number  and  nature  of  the  practical  applications.  With  respect  to 
sequence  the  rule  has  been  followed  that  the  practical  use  of  every 
new  feature  should  be  clearly  set  forth  before  the  abstract  theory  is 
developed.  For  example,  it  will  be  noticed  that  the  definite  uses  of 
each  of  the  natural  functions  are  given  as  soon  as  possible,  that  the 
need  for  logarithmic  computation  follows,  that  thereafter  the  secant 
and  cosecant  assume  a minor  place,  and  that  a wide  range  of  prac- 
tical applications  of  the  right  triangle  awakens  an  early  interest  in 
the  subject.  The  study  of  the  functions  of  larger  angles,  and  of  the 
sum  and  difference  of  two  angles,  now  becomes  necessary  to  further 
progress  in  trigonometry,  after  which  the  oblique  triangle  is  con- 
sidered, together  with  a large  number  of  practical,  nontechnical 
applications. 

The  decimal  division  of  the  degree  is  explained  and  is  used  enough 
to  show  its  value,  but  it  is  recognized  that  this  topic  has,  as  yet, 
only  a subordinate  place.  It  seems  probable  that  the  decimal  frac- 
tion will  in  due  time  supplant  the  sexagesimal  here  as  it  has  in  other 
fields  of  science,  and  hence  the  student  should  be  familiar  with  its 
advantages. 

Such  topics  as  the  radian,  graphs  of  the  various  functions,  the 
applications  of  trigonometry  to  higher  algebra,  and  the  theory  of 
trigonometric  equations  properly  find  place  at  the  end  of  the  course 
in  plane  trigonometry.  They  are  important,  but  their  value  is  best 
appreciated  after  a good  course  in  the  practical  uses  of  the  subject. 

iii 


IV 


PEEFACE 


They  may  be  considered  briefly  or  at  length  as  the  circumstances 
may  warrant. 

The  authors  have  sought  to  give  teachers  and  students  all  the 
material  needed  for  a thorough  study  of  plane  trigonometry,  with 
more  problems  than  any  one  class  will  use,  thus  offering  opportunity 
for  a new  selection  of  examples  from  year  to  year,  and  allowing  for 
the  omission  of  the  more  theoretical  portions  of  Chapters  IX-XII 
if  desired. 

The  tables  have  been  arranged  with  great  care,  every  practical 
device  having  been  adopted  to  save  eye  strain,  all  tabular  material 
being  furnished  that  the  student  will  need,  and  an  opportunity  being 
afforded  to  use  angles  divided  either  sexagesimally  or  decimally,  as 
the  occasion  demands. 

It  is  hoped  that  the  care  that  has  been  taken  to  arrange  all  matter 
in  the  order  of  difficulty  and  of  actual  need,  to  place  the  practical 
before  the  theoretical,  to  eliminate  all  that  is  not  necessary  to  a clear 
understanding  of  the  subject,  and  to  present  a page  that  is  at  the 
same  time  pleasing  to  the  eye  and  inviting  to  the  mind  will  com- 
mend itself  to  and  will  meet  with  the  approval  of  the  many  friends 
of  the  series  of  which  this  work  is  a part. 

GEOKGE  WENTWORTH 
DAVID  EUGENE  SMITH 


CONTENTS 


PLAJ^E  TPIGOJ^OMETEY 

CHAPTER  PAGE 

I.  Trigonometric  Functions  of  Acute  Angles  ....  • 1 

II.  Use  of  the  Table  of  Natural  Functions 27 

III.  Logarithms 39 

IV.  The  Right  Triangle 63 

V.  Trigonometric  Functions  of  any  Angle 77 

VI.  Functions  of  the  Sum  or  the  Difference  of  Two 

Angles 97 

VII.  The  Oblique  Triangle 107 

VIII.  Miscellaneous  Applications 133 

IX.  Plane  Sailing 145 

X.  Graphs  of  Functions  151 

XI.  Trigonometric  Identities  and  Equations 163 

XII.  Applications  of  Trigonometry  to  Algebra 173 

The  Most  Important  Formulas  of  Plane  Trigonometry  . . 185 


V 


Digitized  by  the  Internet  Archive 
in  2016  with  funding  from 
Duke  University  Libraries 


https://archive.org/details/planetrigonometr01went 


PLAE^E  TRIGONOMETRY 


CHAPTER  I 

TRIGONOMETRIC  FUNCTIONS  OF  ACUTE  ANGLES 

1.  The  Nature  of  Arithmetic.  In  aritlunetic  we  study  computation, 
the  working  with  numbers.  We  may  have  a formula  expressed 
in  algebraic  symbols,  such  as  a = Ih,  but  the  actual  computation 
involved  in  applying  such  a formula  to  a particular  case  is  part 
of  arithmetic. 

Arithmetic  enters  into  all  subsequent  branches  of  mathematics.  It  plays 
such  an  important  part  in  trigonometry  that  it  becomes  necessary  to  introduce 
another  method  of  computation,  the  method  which  makes  use  of  logarithms. 

2.  The  Nature  of  Algebra.  In  algebra  we  generalize  arithmetic. 
Thus,  instead  of  saying  that  the  area  of  a rectangle  with  base  4 in. 
and  height  2 in.  is  4 x 2 sq.  in.,  we  express  a general  law  by  saying 
that  a = bh.  Algebra,  therefore,  is  a generalized  arithmetic,  and  the 
equation  is  the  chief  object  of  attention. 

Algebra  also  enters  into  all  subsequent  branches  of  mathematics,  and  its 
relation  to  trigonometry  will  be  found  to  be  very  close. 

3.  The  Nature  of  Geometry.  In  geometry  we  study  the  forms  and 
relations  of  figures,  proving  many  properties  and  effecting  numerous 
constructions  concerning  them. 

Geometry,  like  algebra  and  arithmetic,  enters  into  the  work  in  trigonometry. 
Indeed,  trigonometry  may  almost  be  said  to  unite  arithmetic,  algebra,  and 
geometry  in  one  subject. 

4.  The  Nature  of  Trigonometry.  We  are  now  about  to  begin  another 
branch  of  mathematics,  one  not  chiefly  relating  to  numbers  although 
it  uses  numbers,  not  primarily  devoted  to  equations  although  using 
equations,  and  not  concerned  principally  with  the  study  of  geometric 
forms  although  freely  drawing  upon  the  facts  of  geometry. 

Trigonometry  is  concerned  chiefly  with  the  relation  of  certain 
lines  in  a triangle  {trigon,  "a  triangle,”  + metrein,  "to  measure”)  and 
forms  the  basis  of  the  mensuration  used  in  surveying,  engineering, 
mechanics,  geodesy,  and  astronomy. 

1 


2 


PLANE  TKIGONOMETRY 


5.  How  Angles  are  Measured.  For  ordinary  purposes  angles  can  be 
measured  with  a protractor  to  a degree  of  accuracy  of  about  30'. 

The  student  will  find  it  advantageous  to  use  the  convenient  protractor  fur- 
nished with  this  hook  and  shown  in  the  illustration  below. 


For  work  out  of  doors  it  is  customary  to  use  a transit,  an  instru- 
ment by  means  of  which  angles  can  be  measured  to  minutes.  By 
turning  the  top  of  the  transit  to  the 
right  or  left,  horizontal  angles  can  be 
measured  on  the  horizontal  plate.  By 
turning  the  telescope  up  or  down,  ver- 
tical angles  can  be  measured  on  the 
vertical  circle  seen  in  the  illustration. 

For  astronomical  purposes,  where  great 
care  is  necessary  in  measuring  angles, 
large  circles  are  used. 

The  degree  of  accuracy  required  in  meas- 
uring an  angle  depends  upon  the  nature  of 
the  problem.  We  shall  now  assume  that  we 
can  measure  angles  in  degrees,  minutes,  and 
seconds,  or  in  degrees  and  decimal  parts  of  a degree.  Thus  15°  30'  is  the  same 
as  15.5°,  and  15°  30' 36"  is  the  same  as  15|° of  1°,  or  15.51°. 

The  ancient  Greek  astronomers  had  no  good  symbols  for  fractions.  The  best 
system  they  could  devise  for  close  approximations  was  the  so-called  sexagesimal 
one,  in  which  there  appear  only  the  numerators  of  fractions  whose  denomi- 
nators are  powers  of  60.  This  system  seems  to  have  been  first  suggested  by  the 
Babylonians,  but  to  have  been  developed  by  the  Greeks.  It  is  much  inferior  to 
the  decimal  system  that  was  perfected  about  1600,  but  the  world  still  continues 
to  use  it  for  the  measure  of  angles  and  time.  'Tlie  decimal  division  of  the  angle 
is,  however,  gaining  ground,  and  in  due  time  will  probably  replace  the  more 
cumbersome  one  with  which  we  are  familiar. 

In  this  book  we  shall  use  both  the  ancient  and  modem  systems,  but  with  the 
chief  attention  to  the  former,  since  this  is  still  the  more  common. 


FUNCTIONS  OF  ACUTE  ANGLES 


3 


6.  Functions  of  an  Angle.  In  the  annexed  figure,  if  the  line  AR 
moves  about  the  point  A in  the  sense  indicated  by  the  arrow,  from 
the  position  AX  as  an  initial  position,  it  generates  the  angle  A. 

If  from  the  points  B,B\  B",  . . .,  on  AR,  we  let  fall  the  perpen- 
diculars BC,  B'C,  B”C”,  . . .,  on  AX,  we  form  a series  of  similar 
triangles  ACB,  AC'B',  AC"B",  and  so  on.  The  corresponding  sides 
of  these  triangles  are  proportional.  That  is, 

BC  B'C  B"C" 

AB  ~ AB'  ~ AB" 

BC  _ B^  _ B"C" 

AC  ~ AC  “ AC" 

AB  _ A^  _ AB" 

AC  ~ AC  ~ AC" 

and  similarly  for  the  ratios 

AB  

AC’  Bc’  Ab’ 

each  of  which  has  a series  of  other  ratios  equal  to  it. 


AC  AC 


For  example, 


AB  _ AB'  _ AB" 
^ ~ 'WC'  ~ B"C" ' 


That  is,  these  ratios  remain  unchanged  so  long  as  the  angle  remains 
unchanged,  hut  they  change  as  the  angle  changes. 

Each  of  the  above  ratios  is  therefore  2.  function  of  the  angle  A. 

As  already  learned  in  algebra  and  geometry,  a magnitude  which  depends 
upon  another  magnitude  for  its  value  is  called  a function  of  the  latter  mag- 
nitude.  Thus  a circle  is  a function  of  the  radius,  the  area  of  a square  is  a 
function  of  the  side,  the  surface  of  a sphere  is  a function  of  the  diameter,  and 
the  volume  of  a pyramid  is  a function  of  the  base  and  altitude. 


We  indicate  a function  of  x by  such  symbols  as /(a:),  F(x),  f'(x), 
and  <i>{x),  and  we  read  these  "/of  x,  /-major  of  x, /-prime  of  x,  and 
phi  of  X ” respectively. 

For  example,  if  we  are  repeatedly  using  some  long  expression  like 

+ 3 — 2 -f  7 a;  — 4,  we  may  speak  of  it  briefly  as  f(x).  If  we 

are  using  some  function  of  angle  A,  we  may  designate  this  as /(A). 
If  we  wish  to  speak  of  some  other  function  of  A,  we  may  write  it 
f(A),  F{A),  or  <i>{A). 

In  trigonometry  we  shall  make  much  use  of  various  functions  of  an  angle,  but 
we  shall  give  to  them  special  names  and  symbols.  On  this  account  the  ordinary 
function  symbols  of  algebra,  mentioned  above,  will  not  be  used  frequently  in 
trigonometry,  but  they  will  be  used  often  enough  to  make  it  necessary  that  the 
student  should  understand  their  significance. 


4 


PLAIN’S  TKIGONOMETEY 


7.  The  Six  Functions,  Since  with  a given  angle  A we  may  take 
any  one  of  the  triangles  described  in  § 6,  we 
triangle  ACB,  lettered  as  here  shown. 

It  has  long  been  the  custom  to  letter  in  this  way  the 
hypotenuse,  sides,  and  angles  of  the  first  triangle  con- 
sidered in  trigonometry,  C being  the  right  angle,  and  the 
hypotenuse  and  sides  bearing  the  small  letters  corre- 
sponding to  the  opposite  capitals.  By  the  sides  of  the 
triangle  is  meant  the  sides  a and  b,  c being  called  the 
hypotenuse.  The  sides  a and  b are  also  called  the  legs  of  the  triangle,  par- 
ticularly by  early  writers,  since  it  was  formerly  the  custom  to  represent  the 
triangle  as  standing  on  the  hypotenuse. 


shall  consider  the 
B 


- > — ) - j 7 5 and  - have  the  following  names ; 
c 0 a 0 a 

is  called  the  sine  of  A,  written  sin  A ; 
is  called  the  cosine  of  A , written  cos  A ; 
is  called  the  tangent  of  A,  written  tan  A ; 
is  called  the  cotangent  of  A , written  cot  A ; 


c . 


is  called  the  secant  of  A,  written  sec  A ; 
is  called  the  cosecant  of  A,  written  esc  A. 


The  ratios  -j 
c 

a 

c 

b 

c 

a 

b 

b 

a 

c 

b 

c 

a 

That  is, 

a opposite  side 

sin  A = - — : ) 

c hypotenuse 

a _ opposite  side 
b adjacent  side  ’ 


, b adiacent  side 

cos  A = - = — ; j 

c hypotenuse 

, , b adjacent  side 

COtA=  - = — r- rv-J 

a opposite  side 


_ c _ hypotenuse 
b adjacent  side’ 


, c hypotenuse 
CSC  A = - = — — T7-- 

a opposite  side 


These  definitions  must  be  thoroughly  learned,  since  they  are  the  foundation 
upon  which  the  whole  science  is  built.  The  student  should  practice  upon  them, 
with  the  figure  before  him,  until  he  can  tell  instantly  what  ratio  is  meant  by 
sec  A,  cotA,  sinA,  and  so  on,  in  whatever  order  these  functions  are  given. 

There  are  also  two  other  fimctions,  rarely  used  at  present.  These  are  the 
versed  sine  A = 1 — cos  A,  and  the  co  versed  sine  A = 1 — sin  A.  These  defini- 
tions need  not  be  learned  at  this  time,  since  they  will  be  given  again  when  the 
functions  are  met  later  in  the  work. 


FUNCTIONS  OF  ACUTE  ANGLES 


o 


Exercise  1.  The  Six  Functions 

1.  In  the  figure  of  § 7,  sin  5 = -•  Write  the  other  five  functions 
of  the  angle  B. 

2.  Show  that  in  the  right  triangle  ACB  (§  7)  the  following 
relations  exist : 

sin^  = cos  5,  cos  = sin£,  tan^  = cot£, 
cot4  = tan£,  sec  4 = cscB,  esc  A = seoB. 

State  which  of  the  following  is  the  greater : 

3.  sin^  or  tan^.  5.  sec.I  or  tan^. 

4.  cos^  or  cot^4.  6.  csc^  or  cot^. 

Find  the  values  of  the  six  functions  of  A,  if  a,  b,  c respectively 
have  the  following  values : 

7.  3,  4,  6.  9.  8,  15,  17.  11.  3.9,  8,  8.9. 

8.  5,  12,  13.  10.  9,  40,  41.  12.  1.19,  1.20,  1.69. 

13.  What  condition  must  be  fulfilled  by  the  lengths  of  the  three 
lines  a,  h,  c (fl)  to  make  them  the  sides  of  a right  triangle  ? Show 
that  this  condition  is  fulfilled  in  Exs.  7-12. 


Find  the  values  of  the  six  functions  of  A,  if  a,  b,  c respectively 
have  the  following  values: 


14.  2n,  — 1,  + 1. 

71^  — 1 + 1 

15.  71,  


16. 

17. 


2 77171,  771^  — 71^, 
2 77171 

— ) 771  + 71, 

771  — 71 


771^  + 71^. 
m?  + 71^ 

m — n 


18.  As  in  Ex.  13,  show  that  the  condition  for  a right  triangle  is 
fulfilled  in  Exs.  14-17. 

Given  (F  -\-b^  = find  the  six  functions  of  A when : 

19.  a = h.  20.  a = 2b.  21.  a = ^c. 

Given  a^+b"^  = c^,  fin^  the  six  functions  of  B when : 

22.  a = 24,  b = 143.  24.  a = 0.264,  c = 0.265. 

23.  b = 9.5,  c = 19.3.  25.  b = 2 \/^,  c — p -{■  (I- 


Given  = F,  find  the  six  functions  of  A and  also  the  six 

functions  of  B when : 


26.  a = b — ^2pq. 


27.  a - - + ^,  c ^ + 1. 


6 


PLANE  TPIGONOMETEY 


In  the  right  triangle  A CB,  as  shown  in  § 7: 

28.  Find  the  length  of  side  a if  sind  = and  c — 20.5. 

29.  Find  the  length  of  side  b if  cosd  = 0.44,  and  c = 3.5. 

30.  Find  the  length  of  side  a if  tand  = 3f,  and  h = 2^. 

31.  Find  the  length  of  side  h if  cotA  = 4,  and  a = 1700. 

32.  Find  the  length  of  the  hypotenuse  if  secA  = 2,  and  J = 2000. 

33.  Find  the  length  of  the  hypotenuse  if  cscA=6.4,  and  a = 35.6. 


Find  the  hypotenuse  and  other  side  of  a right  triangle,  given : 

34.  5 = 6,  tanA  = |.  36.  5 = 4,  cscA  = If. 

35.  a = 3.5,  cos  A = 0.5.  37.  5 = 2,  sinA  = 0.6. 

38.  The  hypotenuse  of  a right  triangle  is  2.5  mi.,  sin  A = 0.6,  and 

cos  A = 0.8.  Compute  the  sides  of  the  triangle. 

39.  Construct  with  a protractor  the  angles  20°,  40°,  and  70°; 
determine  their  functions  by  measuring  the  necessary  lines  and 
compare  the  values  obtained  in  this  way  with  the  more  nearly 
correct  values  given  in  the  following  table : 


sin  ^ 

^ COS 

tan 

cot 

sec 

CSC 

20° 

0.342 

0.940 

0.364 

2.747 

1.064 

2.924 

40° 

0.643 

0.766 

i 0.839 

; 1.192 

1.305 

1.556 

70° 

0.940 

0.342 

2.747 

0.364 

2.924 

1.064 

40.  A = 20°,  c = 1. 

41.  A =20°,  c = 4. 

42.  A = 20°,  c = 3.5. 

43.  A = 20°,  c = 4.8. 

44.  A = 20°,  c=7f. 


50.  A = 70°,  c = 2. 

51.  A = 70°,  a=2. 

52.  A = 70°,  5 = 2. 

53.  A = 70°,  a = 25. 

54.  A = 70°,  5 =150. 


Find,  by  means  of  the  above  table,  the  sides  and  hypotenuse  of  a 
right  triangle,  given: 

45.  A = 40°,  c=l. 

46.  A = 40°,  c = 3. 

47.  A=  40°,  c=7. 

48.  A = 40°,  c=10.7. 

49.  A = 40°,  c = 250. 

55.  By  dividing  the  length  of  a vertical  rod  by  the  length  of  its 
horizontal  shadow,  the  tangent  of  the  angle  of  elevation  of  the  sun 
at  that  time  was  found  to  be  0.82.  How  high  is  a tower,  if  the 
length  of  its  horizontal  shadow  at  the  same  time  is  174.3  yd.  ? 

56.  A pin  is  stuck  upright  on  a table  top  and  extends  upward 
1 in.  above  the  surface.  When  its  shadow  is  f in.  long,  what  is  the 
tangent  of  the  angle  of  elevation  of  the  sun  ? How  high  is  a tele- 
graph pole  whose  horizontal  shadow  at  that  instant  is  21  ft.  ? 


FUNCTIONS  OF  ACUTE  ANGLES 


7 


8.  Functions  of  Complementary  Angles.  In  the  annexed  figure  we 
see  that  B is  the  complement  of  A ; that  is,  B = 90°  — A.  Hence, 

sin  ^ = - = cos  B = cos  (90°  — A), 
cos  A =-  = sin  B = sin  (90°  — A), 
tan^  “ ^ ~ ^ ~ 

cot  A = - = tan  B = tan  (90°  — A), 

Q 

sec  ^ ^ = CSC  A = CSC  (90°  — A), 

CSC  A = - = sec  B — sec  (90°  — A). 
a ' 

That  is,  each  function  of  an  acute  angle  is  egual  to  the  co-named 
function  of  the  complementary  angle. 

Co-sine  means  compZemeTit’s  sine,  and  similarly  for  the  other  co-functions. 

It  is  therefore  seen  that  sin  75°  = cos  (90°  — 75°)  = cos  15°,  sec  82°  30'  = 
CSC  (90°  — 82°  30')  = CSC  7°  30',  and  so  on. 

Therefore,  any  function  of  an  angle  between  45°  and  90°  may  he 
found  hy  taking  the  co-named  function  of  the  complementary  angle, 
which  is  between  0°  and  45°. 

Hence,  we  need  never  have  a direct  table  of  functions  beyond  45°.  We  shall 
presently  see  (§  12)  that  this  is  of  great  advantage. 


B 


Exercise  2.  Functions  of  Complementary  Angles 


Express  as  functions  of  the  complementary  angle 


1.  sin  30°. 

2.  cos  20°. 

3.  tan  40°. 

4.  sec  25°. 


5.  sin  50°. 

6.  tan  60°. 

7.  sec  75°. 

8.  CSC  85°. 


9.  sin  60°. 

10.  cos  60°. 

11.  tan  45°. 

12.  sec  45°. 


13.  sin  75°  30'. 

14.  tan  82°  45'. 

15.  sec  68°  15'. 

16.  COS  88°  10'. 


Express  as  functions  of  an  angle  less  than  45°  : 

17.  sin  65°.  20.  cos  52°.  23.  sin  89°. 

18.  tan  80°.  21.  cot  61°.  24.  cos  86°. 

19.  sec  77°.  22.  CSC  78°.  25.  sec  88°. 


26.  sin  77^°. 

27.  cos  82^°. 

28.  tan  88.6°. 


Find  A,  given  the  following  relations: 

29.  90°-A  = A.  31.  90°-A=2A. 

30.  cosA  = sinA.  32.  cosA  = sin2A. 


8 


PLAICE  TEIGONOMETBY 


9.  Functions  of  45°.  The  functions  of  certain  angles,  among  them 
45°,  are  easily  found.  In  the  isosceles  right  triangle  ACB  we  have 
A=i3  = 45°,  and  a = 6.  Furthermore,  since  + = we  have 

2a^  = c^,  a V2  = c,  and  a = \c  V2.  Hence, 

sin  45°  = cos  45°  = ^ ^ 1 ; 

c 2 ’ 

tan  45°  = cot  45°  = ^ = 1; 

sec  45°  = CSC  45°  = ^ = V2. 

a A b 

We  have  therefore  found  all  six  functions  of  45°.  For  purposes  of  computa- 
tion these  are  commonly  expressed  as  decimal  fractions.  Since  V2  = 1.4142  +, 
we  have  the  following  values  : 

sin  45°  = 0.7071,  cos  45°  = 0.7071, 

tan  46°  = 1,  cot  46°  = 1, 

sec  46°  = 1.4142,  esc  45°  = 1.4142. 


10.  Functions  of  30°  and  60°.  In  the  equilateral  triangle  AA'B 
here  shown,  BC  is  the  perpendicular  bisector  of  the  base.  Also, 
b = ^c,  and  a - - Vc^  — 5^  = | c*  = i Hence, 


sec  30°  = CSC  60°  = - = — = ? V3  • 

CSC  30°  = sec  60°  = 7 = 2. 

b 


The  sine  and  cosine  of  30°,  46°,  and  60°  are  easily  remembered,  thus : 
sin  30°  = \ a/I,  sin  46°  = ^ a/2,  sin  60°  = ^ a/S  ; 

cos  30°  = ^ \/3,  cos  45°  = 4 a/^,  cos  60°  = i a/I. 

The  functions  of  other  angles  are  not  so  easily  computed.  The  computation 
requires  a study  of  series  and  is  explained  in  more  advanced  works  on  mathe- 
matics. For  the  present  we  assume  that  the  functions  of  all  angles  have  been 
computed  and  are  available,  as  is  really  the  case. 


FUNCTIO^TS  OF  ACUTE  ANGLES  9 

Exercise  3.  Functions  of  30°,  45°,  and  60° 

Given  Vs  - 1.7320,  express  as  decimal  fractions  the  following  : 


1.  sin  30°.  4.  cot  30°. 

7.  sin  60°.  10.  cot  60°. 

2.  cos  30°.  5.  sec  30°. 

8.  cos  60°.  11.  sec  60°. 

3.  tan  30°.  6.  esc  30°. 

9.  tan  60°.  12.  esc  60°. 

Write  the  ratios  of  the  following,  simplifying  the  results : 

13.  sin  45°  to  sin  30°. 

19.  sin  30°  to  sin  60°. 

14.  cos  45°  to  cos  30°. 

20.  cos  30°  to  cos  60°. 

15.  tan  45°  to  tan  30°. 

21.  tan  30°  to  tan  60°. 

16.  cot  45°  to  cot  30°. 

22.  cot  30°  to  cot  60°. 

17.  sec  45°  to  sec  30°. 

23.  sec  30°  to  sec  60°. 

18.  CSC  45°  to  CSC  30°. 

24.  CSC  30°  to  CSC  60°.  ly' 

Express  as  functions  of  angles  less 

than  46° : 

25.  sin  62°  17'  40". 

29.  sin  75.8°. 

26.  tan  75°  28'  35". 

30.  cos  82.75°. 

27.  sec  87°  32'  51". 

31.  tan  68.82°. 

28.  cos  88°  0'  27". 

32.  sec  85.95°. 

Find  A,  given  the  following  relations: 

33.  90°  - 45°  - ^A. 

38.  cos  A = sin  (45°  — 

34.  90°  - \A=A. 

39.  cot|^A  = tanA. 

35.  45°  +A=90°  —A. 

40.  tan  (45°  + A)  = cot  A. 

36.  90°  — 4.A=A. 

41.  cos  4 A = sin  A. 

37.  90°— A = nA. 

42.  cot  A = tanwA. 

43.  By  wbiat  must  siu  45°  be  multiplied  to  equal  tan  30°  ? 

44.  By  ’what  must  sec  45°  be  multiplied  to  equal  esc  30°  ? 

45.  By  what  must  cos  45°  be  multiplied  to  equal  tan  60°  ? 

46.  By  what  must  esc  60°  be  divided  to  equal  tan  45°  ? 

47.  By  what  must  esc  30°  be  divided  to  equal  tan  30°  ? 

48.  What  is  the  ratio  of  sin  45°  sec  45°  to  cos  60°  ? 

49.  What  is  the  ratio  of  cos  45°  esc  45°  to  cos  30°  esc  30°  ? 

50.  What  is  the  ratio  of  sin  45°  sin  30°  to  cos  45°  cos  30°  ? 

51.  What  is  the  ratio  of  tan  30°  cot  30°  to  tan  60°  cot  60°  ? 
62.  From  the  statement  tan  30°  = ^ Vs  find  cot  60°. 


10 


PLANE  TRIGONOMETEY 


11.  Values  of  the  Functions.  The  values  of  the  functions  have 
been  computed  and  tables  constructed  giving  these  values.  One 
of  these  tables  is  shown  on  page  11  and  will  suffice  for  the  work 
required  on  the  next  few  pages. 

This  table  gives  the  values  of  the  functions  to  four  decimal  places  for  every 
degree  from  0°  to  90°.  All  such  values  are  only  approximate,  the  values  of  the 
functions  being,  in  general,  incommensurable  with  unity  and  not  being  ex- 
pressible by  means  of  common  fractions  or  by  means  of  decimal  fractions  with 
a finite  number  of  decimal  places. 

12.  Arrangement  of  the  Table.  As  explained  in  § 8,  cos  45°  = sin  45°, 
cos  46°  = sin  44°,  cos  47°  = sin  43°,  and  so  on.  Hence  the  column 
of  sines  from  0°  to  45°  is  the  same  as  the  column  of  cosines  from 
45°  to  90°.  Therefore 

In  finding  the  functions  of  angles  from  0°  to  45°  read  from  the  top 
down  ; in  finding  the  functions  of  angles  from  45°  to  90°  read  from 
the  bottom  up. 

Exercise  4.  Use  of  the  Table 

From  the  table  on  page  11  find  the  values  of  the  following : 


1. 

sin 

5°. 

9. 

COS 

6°. 

17. 

cot  5°. 

25. 

sec  0°. 

2. 

sin 

14°. 

10. 

sin 

84°. 

18. 

tan  85°. 

26. 

CSC  90°, 

3. 

sin 

21°. 

11. 

cos 

14°. 

19. 

cot  11°. 

27. 

sec  15°. 

4. 

sin 

30°. 

12. 

sin 

76°. 

20. 

tan  79°. 

28. 

esc  75°. 

6. 

cos 

85°. 

13. 

cos 

24°. 

21. 

tan  21°. 

29. 

CSC  12°. 

6. 

cos 

76°. 

14. 

sin 

66°. 

22. 

cot  69°. 

30. 

sec  78°. 

7. 

cos 

69°. 

15. 

cos 

35°. 

23. 

tan  45°. 

31. 

CSC  44°. 

8. 

cos 

60°. 

16. 

sin 

55°. 

24. 

cot  45°. 

32. 

sec  46°. 

33.  Find  the  difference  between  2 sin  9°  and  sin  (2  x 9°). 

34.  Find  the  difference  between  3 tan  5°  and  tan  (3  x 5°). 

36.  Which  is  the  larger,  2 sec  10°  or  sec  (2  x 10°)? 

36.  Which  is  the  larger,  2 cscl0°  or  esc (2  x 10°)? 

37.  Which  is  the  larger,  2 cos  15°  or  cos  (2  x 15°)? 

38.  Compare  3 sin  20°  with  sin  (3  x 20°);  with  sin  (2  x 20°). 

39.  Compare  3 tan  10°  with  tan  (3  x 10°);  with  tan  (2  x 10°). 

40.  Compare  3 cos  10°  with  cos  (3  x 10°) ; with  cos  (2  x 10°). 

41.  Is  sin  (10°  + 20°)  equal  to  sin  10°  + sin  20°  ? 

42.  When  the  angle  is  increased  from  0°  to  90°  which  of  the  six 
functions  are  increased  and  which  are  decreased  ? 


FUNCTIONS  OF  ACUTE  ANGLES 


11 


Table  of  Trigonometric  Functions  for  evert  Degree 
FROM  0°  TO  90° 


Angle 

sin 

cos 

tan 

cot 

sec 

CSC 

0° 

.0000 

1.0000 

.0000 

00 

1.0000 

00 

90° 

1° 

.0175 

.9998 

.0175 

57.2900 

1.0002 

57.2987 

89° 

2° 

.0349 

.9994 

.0349 

28.6363 

1.0006 

28.6537 

88° 

3° 

.0523 

.9986 

.0524 

19.0811 

1.0014 

19.1073 

87° 

4° 

.0698 

.9976 

.0699 

14.3007 

1.0024 

14.3356 

86° 

5° 

.0872 

.9962 

.0875 

11.4301 

1.0038 

11.4737 

85° 

6^ 

.1045 

.9945 

.1051 

9.5144 

1.0055 

9.5668 

84° 

7° 

.1219 

.9925 

.1228 

8.1443 

1.0075 

8.2055 

83° 

8° 

.1392 

.9903 

.1405 

7.1154 

1.0098 

7.1853 

82° 

90 

.1564 

.9877 

.1584 

6.3138 

1.0125 

6.3925 

81° 

0 

© 

.1736 

.9848 

.1763 

5.6713 

1.0154 

5.7588 

80° 

11° 

.1908 

.9816 

.1944 

5.1446 

1.0187 

5.2408 

79° 

12° 

.2079 

.9781 

.2126 

4.7046 

1.0223 

4.8097 

78° 

13° 

.2250 

.9744 

.2309 

4.3315 

1.0263 

4.4454 

77° 

14° 

.2419 

.9703 

.2493 

4.0108 

1.0306 

4.1336 

76° 

15° 

.2588 

.9659 

.2679 

3.7321 

1.0353 

3.8637 

75° 

16° 

.2756 

.9613 

.2867 

3.4874 

1.0403 

3.6280 

74° 

17° 

.2924 

.9563 

.3057 

3.2709 

1.0457 

3.4203 

73° 

18° 

.3090 

.9511 

.3249 

3.0777 

1.0515 

3.2361 

72° 

19° 

.3256 

.9455 

.3443 

2.9042 

1.0576 

3.0716 

71° 

20° 

.3420 

.9397 

.3640 

2.7475 

1.0642 

2.9238 

70° 

21° 

.3584 

.9336 

.3839 

2.6051 

1.0711 

2.7904 

69° 

22° 

.3746 

.9272 

.4040 

2.4751 

1.0785 

2.6695 

68° 

23° 

.3907 

.9205 

.4245 

2.3559 

1.0864 

2.5593 

67° 

24° 

.4067 

.9135 

.4452 

2.2460 

1.0946 

2.4586 

66° 

25° 

.4226 

.9063 

.4663 

2.1445 

1.1034 

2.3662 

65° 

26° 

.4384 

.8988 

.4877 

2.0503 

1.1126 

2.2812 

64° 

27° 

.4540 

.8910 

.5095 

1.9626 

1.1223' 

2.2027 

63° 

28° 

.4695 

.8829 

.5317 

1.8807 

1.1326 

2.1301 

62° 

29° 

.4848 

.8746 

.5543 

1.8040 

1.1434 

2.0627 

61° 

30° 

.5000 

.8660 

.5774 

1.7321 

1.1547 

2.0000 

60° 

31° 

.5150 

.8572 

.6009 

1.6643 

1.1666 

1.9416 

59° 

32° 

.5299 

.8480 

.6249 

1.6003 

1.1792 

1.8871 

58° 

33° 

.5446 

.8387 

.6494 

1.5399 

1.1924 

1.8361 

57° 

34° 

.5592 

.8290 

.6745 

1.4826 

1.2062 

1.7883 

56° 

35° 

.5736 

.8192 

.7002 

1.4281 

1.2208 

1.7434 

55° 

36° 

.5878 

.8090 

.7265 

1.3764 

1.2361 

1.7013 

54° 

37° 

.6018 

.7986 

.7536 

1.3270 

1.2521 

1.6616 

53° 

38° 

.6157 

.7880 

.7813 

■ 1.2799 

1.2690 

1.6243 

52° 

39° 

.6293 

.7771 

.8098 

1.2349 

1.2868 

1.5890 

51° 

40° 

.6428 

.7660 

.8391 

1.1918 

1.3054 

1.5557 

50° 

41° 

.6561 

.7547 

.8693 

1.1504 

1.3250 

1.5243 

49° 

42° 

.6691 

.7431 

.9004 

1.1106 

1.3456 

1.4945 

48° 

43° 

.6820 

.7314 

.9325 

1.0724 

1.3673 

1.4663 

47° 

44° 

.6947 

.7193 

.9657 

1.0355 

1.3902 

1.4396 

46° 

45° 

.7071 

.7071 

1.0000 

1.0000 

1.4142 

1.4142 

45° 

COS 

sin 

cot 

tan 

CSC 

sec 

Angle 

12 


PLANE  TKIGONOMETRY 


13.  Reciprocal  Functions.  Considering  the  definitions  of  the  six 
functions,  we  see  that,  since 


sin  A = - > 
c 

, c 

CSC  A = - > 
a 


cos  A - 


sec  A = - ) 
0 


tanA  - ^ > 
0 


cot  A = - 1 
a 


The  sine  is  the  reciprocal  of  the  cosecant,  the  cosine  is  the  reciprocal 
of  the  secant,  and  the  tangent  is  the  reciprocal  of  the  cotangent. 

That  is,  ^ ^ 

sinA  = -1  cosA  = r?  tanA=  ^ 


cscA  - - 


CSC  A 

1 


sec  A: 


sec  A 

1 


cot  A = 


cot  A 
1 


sin  A ' cos  A tanA 

HLen.Qe_sinA  esc  A = 1,  cosA  secA  = 1.  and  tanA  cot  A = J.  For  example, 
from  the  table  on  page  11  we  find  sin  27°  esc  27°  thus  T 
sin  27°  = 0.4540. 

CSC  27°  = 2.2027. 

Therefore  sin  27°  csc27°  = 0.4540  x 2.2027 

= 1.00002580,  or  approximately  1. 

We  have  shown  that  sin  A esc  A = 1 exactly,  but  the'numbers  given  in  the 
table  are,  as  before  stated,  correct  only  to  four  decimal  places. 


Exercise  5.  Use  of  the  Table 

Using  the  values  .given  in  the  table  on  page  11,  show  as  above  that 
the  following  are  reciprocals : 

1.  sin  30°,  CSC  30°.  4.  sinl0°,  cscl0°.  7.  sin  75°,  esc  75°. 

2.  sin  25°,  CSC  25°.  5.  tan  10°,  cot  10°.  8.  cos  75°,  sec  75°. 

3.  cos  35°,  sec  35°.  6.  cosl0°,  secl0°.  9.  tan  75°,  cot  75°. 

10.  From  the  table  show  that  the  ratio  of  sin  20°  esc  20°  to  tan  50° 
cot  50°  is  1. 

11.  Similarly,  show  that  cos  40°  sec  40°  : tan  70°  cot  70°  =1. 

In  the  right  triangle  A CB,  as  shown  in  § 7 : 

12.  Find  the  length  of  side  a.  if  A = 30°,  and  c = 75.2. 

13.  Find  the  length  of  side  a if  A = 45°,  and  c = 1.414. 

14.  Find  the  length  of  side  h \i  A = 30°,  and  c = 115.47. 

15.  Find  the  length  of  side  a if  A = 60°,  and  b — 34.64. 

16.  Find  the  length  of  side  5 if  A = 60°,  and  c = 25.72. 

17.  Find  the  length  of  side  a if  A = 30°,  and  c = 45.28. 


FUNCTIONS  OF  ACUTE  ANGLES 


13 


14.  Other  Relations  of  Functions.  Since,  from  the  figure  in  § 7, 


a*  -f-  z=  c^,  we  have 


or 


- + - = 1 

sin^A  + cos*A  = 1 . 


It  is  customary  to  write  sin^A  for  (sinA^  and  similarly  for  the  other 
functions.  

This  formula  is  one  of  the  most  important  in  trigonometry  and 
should  be  memorized.  From  it  we  see  that 


sin  A - Vl  — cos^A, 


cos 


A = Vl  — sin^A. 


Furthermore,  since  tan  A = sin  A = ->  and  cos  A = -?  it  follows 

he  c 

, sinA 

tan  A = 

cos  A 

This  is  also  an  important  formula  to  be  memorized.  From  it  we  see  that 
tan  A cos  A = sinA,  and,  in  general,  that  we  can  find  any  one  of  the  functions, 
sine,  cosine,  or  tangent,  given  the  other  two. 

Furthermore,  from  the  same  equation  we  see  that 


. tt/ 

1 = 


Hence  we  see  that 


1 + tan^A  = secM. 


In  a similar  manner  we  may  prove  that  14-  — =:—;  whence  we 
have  the  formula  i .p  cotU  = csc^A. 


These  two  formulas  should  be  memorized. 

From  these  formulas  the  following  relations  can  easily  be  deduced : 
sin  X = cos  X tana;  = cos  a;/cot  x = tanx/sec  x. 
cos  X = cot  X sin  x = cot  x/ese  x = sin  x/tanx. 
tanx  = sin  x sec  x = sin  x/cos  x = sec  x/ese  x. 
cot  X - - CSC  X cos  X = CSC  x/sec  x = cos  x/sin  x. 
sec  X = tanx  esc  x = tan  x/sin  x = esc  x/cot  x. 

CSC  X = sec  X cot  x = sec  x/tanx  = cot  x/cos  x. 


It  is  often  convenient  to  recall  thesg_relations,  and  this  can  be  done  by  the 
aid  of  a simple  mnemonic : .^''^tan  x 

sin  X sec  x ' 

,cosx  esex; 

cotx 

In  the  above  diagram,  any  function  is  equal  to  the  product  of  the  two  adjacent 
functions,  or  to  the  quotient  of  either  adjacent  function  divided  try  the  one  beyond  it. 


14 


PLANE  TRIGONOMETEY 


15.  Practical  Use  of  the  Sine.  Since  by  definition  we  iiaYe 

- = sinA, 
c 

we  see  that  a = c sin  A. 

We  might  also  derive  the  equation 

a 

c 

sin  A 

But  since  — - — = esc  A (§  13),  it  is  easier  at  present  to  use 
sin  A 


c = a cscA, 

and  this  will  be  considered  when  we  come  to  study  the  cosecant. 


1.  Given  c = 38  and  A = 40°,  find  a. 

As  above,  a = c sin  A. 

From  the  table,  sin  40°  = 0.6428 

and  c = 38 

51424 
19  284 

c sin  A = 24.4264 

But  since  the  table  on  page  11  gives  only  the  first  four  figures  of  sin  40'’,  we 
can  expect  only  the  first  four  figures  of  the  result  to  be  correct.  We  therefore 
say  that  a = 24.43  — . If  the  third  decimal  place  were  less  than  5,  the  value 
of  a would  be  written  24.42  +. 

Some  check  should  always  be  applied  to  the  result.  In  this  case  we  may 
proceed  as  follows  : 24.4264  38  = 0.6428,  which  is  sin  40°. 


2.  Given  c = 10  and  a = 6.293,  find  A. 

a 


Since 
we  have 


c 

6.293 

10 


= sin  A, 

= 0.6293  = sinA. 


Looking  in  the  table  we  see  that 

0.6293  = sin  39°; 

whence  A - 39°. 


3.  Given  a — 4.68^  and  A = 22°,  find  c. 

As  stated  above,  c may  be  found  from  the  formula  a = c sin  A by 
using  a and  sin  A,  although  we  shall  later  use  the  cosecant  for  this 
purpose.  Substituting  the  given  values,  we  have 

4.684  = sin  22°, 
or  4.6825  = 0.3746  c. 

Dividing  by  0.3746,  12.5  = c. 

What  check  should  be  applied  here  and  in  Ex.  2 ? 


FUNCTIONS  OF  ACUTE  ANGLES 


15 


Exercise  6.  Use  of  the  Sine 


Find  a to  four  figures,  given  the  following : 

1.  c = 10,  10’.  3.  c = 58,  A = 45°. 

2.  c = 15,  A = 15°.  4.  c = 75,  A = 50°. 


Find  A,  given  the  following : 
b.  c = 10,  a = 2.079.  7.  c = 2,  a =1.2586. 

6.  c = 20,  a = 6.840.  8.  c = 50,  a — 34.1. 

9.  A 50-foot  ladder  resting  against  the 
side  of  a house  reaches  a point  25  ft.  from 
the  ground.  What  angle  does  it  make  with 
the  ground  ? 

In  all  such  cases  the  ground  should  he  con.sidered  level  and  the  side  of  the 
building  should  be  considered  vertical  unless  the  contrary  is  expressly  stated. 


10.  From  the  top  of  a rock  a cord  is 
stretched  to  a point  on  the  ground,  making 
an  angle  of  40°  with  the  horizontal  plane. 
The  cord  is  84  ft.  long.  Assuming  the  cord 
to  be  straight,  how  high  is  the  rock? 


11.  Find  the  side  of  a regular  decagon  in- 
scribed in  a circle  of  radius  7 ft. 

What  is  the  central  angle?  What  is  half  of  this 
angle  ? Find  BG  and  double  it.  By  this  plan  we  can 
find  the  perimeter  of  any  inscribed  regular  polygon, 
given  the  radius  of  the  circle.  In  this  way  we  could 
approximate  the  value  of  tt.  For  example,  we  see  that  the  semiperimeter  of  a 
polygon  of  90  sides  in  a unit  circle  is  90  x sin  2°,  or  90  x 0.0349,  or  3.141. 


12.  The  edge  of  the  Great  Pyramid  is 
609  ft.  and  makes  an  angle  of  52°  with  the 
horizontal  plane.  What  is  the  height  of  the 
pyramid  ? 


13.  Wishing  to  measure  BC,  the  length  of  a 
pond,  a surveyor  ran  a line  CA  at  right  angles 
to  BC.  He  measured  AB  and  Z.A,  finding 
that  A5=  928  ft.,  and  A = 29°.  Find  the 
length  of  BC. 

In  practical  surveying  we  would  probably  use  an  oblique  triangle,  although 
the  work  as  given  here  is  correct.  The  oblique  triangle  is  considered  later. 


16 


PLANE  TRIGONOMETEY 


16.  Practical  Use  of  the  Cosine.  Since  by  definition  we  have 

h 

- = eosA, 
c 

we  see  that  h = c cos  A. 

1.  Given  c = 28  and  A = 46°,  find  b. 

From  the  table,  cos  46°  = 0.6947 

and  e—  28 

5 5576 
13  894 
19.4516 

Hence,  to  four  figures,  h = 19.45. 

2.  Given  c = 2 and  b = 1.9022,  find  A. 

Since  - = cosA, 

c 

we  have  1.9022  ^ 2 = 0.9511  = cos  A. 

From  the  table,  0.9511  = cos  18°. 

Hence  A = 18°. 

What  check  should  be  applied  here  and  in  Ex.  1 ? 

Exercise  7.  Use  of  the  Cosine 


Find  h to  four  figures,  given  the  following  : 


1.  c = 11,  A = 10°. 

6. 

c = 2.8, 

O * 
00 

II 

2.  c 14,  A = 16°. 

7. 

c = 9.7, 

A = 52°. 

3.  c = 28,  A = 24°. 

8. 

c = 11.2, 

A = 58°. 

4.  c = 41,  A =3  39°. 

9. 

c = 12.5, 

A = 67°. 

5.  c = 75,  A = 42°. 

10. 

c = 28.25; 

, A = 75°. 

Find  A,  given  the  following : 

11.  c = 10,  6 = 9.848. 

16. 

c = 600, 

6 = 205.2. 

12.  c = 20,  6 = 19.126. 

17. 

c = 200, 

6 = 117.56. 

13.  c = 40,  6 = 35.952. 

18. 

c = 187, 

6 = 93^. 

14.  c = 17.6,  6 = 8.8. 

19. 

c = 300, 

6 = 102|. 

15.  c = 500,  6 = 227. 

20. 

c = 1000, 

6 = 1044. 

21.  A flagstaff  breaks  off  22  ft.  from  the  top  and,  the  parts  still 
holding  together,  the  top  of  the  staff  reaches  the  earth  11  ft.  from 
the  foot.  What  angle  does  it  make  with  the  ground  ? 


FUNCTIONS  OF  ACUTE  ANGLES 


17 


22.  Wishing  to  measure  the  length  of  a pond, 
a class  constructed  a right  triangle  as  shown  in 
the  figure.  If  = 640  ft.  and  A ---  50°,  required 
the  distance  AC. 

23.  In  the  same  figure  what  is  the  length  of 
AC  when  AB  = 600  ft.  and  A = 40°  ? 

24.  In  the  same  figure,  it  AC  = 731.4  ft.  and  AB=  1000  ft.,  what 
is  the  size  of  angle  A ? 

26.  A regular  hexagon  is  inscribed  in  a circle  of 
radius  9 in.  How  far  is  it  from  the  center  to  a side  ? 

Having  found  this  distance,  the  apothem,  and  knowing 
that  a side  of  the  regular  hexagon  equals  the  radius,  we 
can  find  the  area,  as  required  in  Ex.  26. 

26.  What  is  the  area  of  a regular  hexagon  inscribed  in  a circle  of 
radius  8 in.  ? 

27.  A ship  sails  northeast  8 mi.  It  is  then  how  many  miles  to  the 
east  of  the  starting  point  ? 

Northeast  is  46°  east  of  north.  In  all  such  cases  in  plane  trigonometry  the 
figure  is  supposed  to  be  a plane.  For  long  distances  it  would  be  necessary  to 
consider  a spherical  triangle. 


A' 


28.  Some  16-foot  roof  timbers  make  an  angle  of 
30°  with  the  horizontal  in  an  A-shaped  roof,  as 
shown  in  the  figure.  Find  A A the  span  of  the  roof. 

29.  An  equilateral  triangle  is  inscribed  in  a circle  of  radius  12  in. 
How  far  is  it  from  the  center  to  a side  ? 

30.  A crane  AB,  30  ft.  long,  makes  an  angle 
of  X degrees  with  the  horizontal  line  AC.  Find 
the  distance  A C when  x = 20-,  when  a;  = 45 ; when 
a;  = 65  ; when  a;  = 0 ; when  x = 90. 

31.  In  Ex.  30.  what  angle  does  the  crane  make  with  the  horizontal 
when  AC  = 15  ft. ? when  ^ C = 30  ft. ? 

32.  The  square  AN,  of  which  the  side  is  200  ft., 
is  inscribed  in  the  square  CM.  AC  is  181.26  ft. 

Required  the  angles  that  the  sides  of  the  small 
square  make  with  the  large  one. 

G 

33.  In  Ex.  32  find  the  required  angles  when 

^5  = 15  in.  and  BC  = 1\  in.;  when  AB  = 20  in.  andRC  = 10.3  in. 

84.  The  edge  of  the  Great  Pyramid  is  609  ft.,  and  it  makes  an  angle 
of  52°  with  the  horizontal  plane.  What  is  the  diagonal  of  the  base  ? 


18 


PLANE  TKIGONOMETKY 


17.  Practical  Use  of  the  Tangent.  Since  by  definition  we  have 

V = tanA, 

0 

we  see  that  a = b tan  A. 

Given  h = 12  and  A = 35°,  find  a. 

From  the  table,  tan  35°  = 0.7002 

h=  12 
1 4004 
7 002 
8.4024 

Hence,  to  four  figures,  a = 8.402. 

The  figures  1,  2,  — ,9  are  often  spoken  of  as  significant  figures.  In  8.402  the 
zero  is,  however,  looked  upon  as  a significant  figure,  but  not  in  a case  like 
12,550.  The  first  four  significant  figures  in  0.6705067  are  6705. 

18.  Angles  of  Elevation  and  Depression.  The  angle  of  elevation,  or 
the  angle  of  depression,  of  an  object  is  the  angle  which  a line  from 
the  eye  to  the  object  makes 
with  a horizontal  line  in  the 
same  vertical  plane. 

Thus,  if  the  observer  is  at  0,  x 
is  the  angle  of  elevation  of  B,  and 
y is  the  angle  of  depression  of  G. 

In  measuring  angles  with  a 
transit  the  height  of  the  instru- 
ment must  always  be  taken  into  account.  In  stating  problems,  however,  it  is  not 
convenient  to  consider  this  every  time,  and  hence  the  angle  is  supposed  to  be 
taken  from  the  level  on  which  the  instrument  stands,  unless  otherwise  stated. 

1.  From  a point  5 ft.  above  the  ground  and  150  ft.  from  the  foot 
of  a tree  the  angle  of  elevation  of  the  top  is  observed  to  be  20°. 
How  high  is  the  tree  ? 

W e have  a = 6 tan  A 

= 150  tan  20° 

= 150  X 0.3640 
= 54.6. 

Hence  the  height  of  the  tree  is  54.6  ft.  -f-  5 ft.,  or  59.6  ft. 

2.  From  a point  A on  a cliff  60  ft.  high,  including  the  instrument, 
the  angle  of  depression  of  a boat  A on  a lake  is  observed  to  be  25°. 
How  far  is  the  boat  from  C,  the  foot  of  the  cliff  ? 

We  have  Z AA  C= 65°.  Hence  AC= 60  tan  65°.  From  the 
table,  tan  65°  = 2.1445.  Hence  AC  = 60  x 2.1445  = 128.67.  £ 


B 


FUNCTIONS  OF  ACUTE  ANGLES 


19 


Exercise  8.  Use  of  the  Tangent 
Find  a to  four  significant  figures,  given  the  following : 


1.  b = 37,A=^  18°. 

6. 

5 = 4.8,  A =51°. 

2.  5 = 26,  A = 23°. 

7. 

5 = 9.6,  A =57°. 

3.  5 = 48,  A = 31°. 

8. 

5 = 23.4,  A = 62°. 

4.  5 = 62,  A = 36°. 

9. 

5 = 28.7,  A = 75°. 

6.  5 = 98,  A = 45°. 

10. 

5 = 39.7,  A = 85°. 

Find  A,  given  the  following : 

11.  a = 6,  5 = 6. 

14. 

a =13.772,  5 = 40. 

12.  a = 0.281,  5 = 2. 

15. 

a = 2.424,  5 = 6. 

13.  a = 4.752,  5 = 30. 

16. 

a = 20.503,  5 =10. 

17.  A man  standing  120  ft.  from  the  foot  of  a church  spire  finds 
that  the  angle  of  elevation  of  the  top  is  50°.  If  his  eye  is  5 ft.  8 in. 
from  the  ground,  what  is  the  height  of  the  spire  ? 

18.  When  a flagstaff  55.43  ft.  high  casts  a shadow  100  ft.  long 
on  a horizontal  plane,  what  is  the  angle  of  elevation  of  the  sun  ? 

19.  A ship  S is  observed  at  the  same  instant 
from  two  lighthouses,  L and  L',  3 mi.  apart. 

Z.L'LS  is  found  to  be  40°  and  Z.LL'S  is  found  to 
he  90°.  What  is  the  distance  of  the  ship  from  L'  ? 

What  is  its  distance  from  L ? 

20.  From  the  top  of  a rock  which  rises  vertically,  including  the 
instrument,  134  ft.  above  a river  bank  the  angle  of  depression  of 
the  opposite  bank  is  found  to  be  40°.  How  wide  is  the  river  ? 

21.  An  A-shaped  roof  has  a span^ff'of  24  ft.  The 
ridgepole  A is  12 ft.  above  the  horizontal  line  AA'. 

What  angle  does  AR  make  with  AA'  ? with  RA'  ? 
with  the  perpendicular  from  R on  AA'? 

22.  The  foot  of  a ladder  is  17  ft.  6 in.  from  a wall,  and  the  ladder 
makes  an  angle  of  42°  with  the  horizontal  when  it  leans  against 
the  wall.  How  far  up  the  wall  does  it  reach  ? 

23.  A post  subtends  an  angle  of  7°  from  a point  on  the  ground 
50  ft.  away.  What  is  the  height  of  the  post  ? 

24.  The  diameter  of  a one-cent  piece  is  f in.  If  the  coin  is  held 
so  that  it  subtends  an  angle  of  40°  at  the  eye,  what  is  its  distance 
from  the  eye  ? 


20 


PLANE  TRIGONOMETKY 


19.  Practical  Use  of  the  Cotangent.  Since  by  definition  we  have 


we  see  that 


h 

- = cotA, 
a 

b = a cot  A. 


For  example,  given  a.  = 71  and  A = 28®, 
find  b. 

From  the  table,  cot  28°  = 1.8807 

and  a=  71 

1 8807 
131  649 
133.6297 

Hence,  to  four  significant  figures,  b = 133.5. 
What  check  should  be  applied  in  this  case  ? 


Exercise  9.  Use  of  the  Cotangent 


Find  h to  four  significant  figures,  given  the  following : 


1.  a = 29,  A = 48®. 

2.  a = 38,  A = 72°. 

3.  a = 56,  A = 19°. 

4.  a = 72,  A = 40°. 


6.  a = 425,  A = 38®. 
6.  a = 19^,  A = 36°. 
1.  a = 24.8,  A = 43°. 
8.  a = 256.8,  A = 75®. 


Find  A,  given  the  following : 

9.  a=12,b  = 72.  10.  a = 60,  =128.67. 


11.  How  far  from  a tree  50  ft.  high  must  a person  lie  in  order  to 
see  the  top  at  an  angle  of  elevation  of  60°  ? 

12.  From  the  top  of  a tower  300  ft.  high,  in- 
cluding the  instrument,  a point  on  the  ground 
is  observed  to  have  an  angle  of  depression  of 
35°.  How  far  is  the  point  from  the  tower  ? 


300 


13.  From  the  extremity  of  the  shadow  cast  by  a church  spire 
150  ft.  high  the  angle  of  elevation  of  the  top  is  53®.  What  is  the 
length  of  the  shadow  ? 

14.  A tree  known  to  be  50  ft.  high,  stand- 
ing on  the  bank  of  a stream,  is  observed 
from  the  opposite  bank  to  have  an  angle  of 
elevation  of  20®.  The  angle  is  measured 
on  a line  5 ft.  above  the  foot  of  the  tree.  How  wide  is  the  stream  ? 


FUNCTIONS  OF  ACUTE  ANGLES 


21 


20.  Practical  Use  of  the  Secant.  Since  by  definition  we  have 

7 = secA, 

0 

we  see  that  c = h sec  A. 

For  example,  given  5 = 15  and  A = 30°,  find  c. 

From  the  table,  sec  30°  = 1.1547 

and  h = 15 

5 7735 
11  547 
17.3205 

Hence,  to  four  significant  figures,  c = 17.32. 


Exercise  10.  Use  of  the  Secant 


Find  c to  four  significant  figures,  given  the  following  : 

1.  b = 36,A=  27°.  4.  6 = 22^,  A = 48°. 

2.  6 = 48,  A = 39°.  5.6  = 33.4,  A = 53°. 

3.  6 = 74,  A = 43°.  6.6  = 148.8,  A = 64°. 


Find  A,  given  the  following : 


7.  6 = 10,  c = 131.  8.  6 = 17.8, 

9.  A ladder  rests  against  tbe  side  of  a build- 
ing, and  makes  an  angle  of  28°  with  tbe  ground. 
Tbe  foot  of  tbe  ladder  is  20  ft.  from  tbe  building. 
How  long  is  tbe  ladder  ? 

10.  From  a point  50  ft.  from  a bouse  a wire 
window  so  as  to  make  an  angle  of  30°  witb  tbe 
tbe  length  of  tbe  wire,  assuming  it  to  be  straight. 

11.  In  measuring  the  distance  A£  a surveyor 
ran  tbe  line  AC,  making  an  angle  of  50°  witb  AB, 
and  tbe  line  BC  perpendicular  to  AC.  He  meas- 
ured AC  and  found  that  it  was  880  ft.  Eeqnired 
tbe  distance  AB. 


e = 35.6. 


horizontal.  Find 


12.  From  tbe  extremity  of  tbe  shadow  cast  by  a tree  tbe  angle  of 
elevation  of  tbe  top  is  47°.  Tbe  shadow  is  62  ft.  6 in.  long.  How 
far  is  it  from  tbe  top  of  tbe  tree  to  tbe  extremity  of  tbe  shadow  ? 


13.  Tbe  span  of  this  roof  is  40  ft.,  and  tbe  roof 
timbers  AB  make  an  angle  of  40°  witb  tbe  hori- 
zontal. Find  tbe  length  of  AB. 


22 


PLAJSTE  TRIGONOMETRY 


21.  Practical  Use  of  the  Cosecant.  Since  by  definition  we  have 


— ~ GSC^, 
a 


we  see  that 


c = a csc^. 

For  example,  given  a = 22  and  A = 35°, 
find  c. 

From  the  table.  esc  35°  = 1.7434 
and  a=  22 

3 4868 
34  868 
38.3548 


Hence,  to  four  significant  figures,  c = 38.35. 

GUck.  Since  - = sin^,  22  38.35  = 0.5736  = sin 35°. 

c 


Exercise  11.  Use  of  the  Cosecant 

Find  c to  four  significant  figures,  given  the  following : 

1.  a = 24,  d = 29°.  4.  a = 56^,  A = 61°. 

2.  a = 36,  ^ = 41°.  5.  a = 75.8,  A = 69°. 

3.  a = 56,A  = 44°.  6.  a = 146.9,  A = 74°. 

Find  A,  given  the  following  : 

7.  a = 10,  c = 11.126.  9.  a = 5^,c  = 6.0687. 

8.  a = 13,  c = 27.6913.  10.  a = 75,c  = 106.065. 

11.  Seen  from  a point  on  the  ground  the  angle  of  elevation  of  an 
aeroplane  is  64°.  If  the  aeroplane  is  1000  ft.  above  the  ground,  how 
far  is  it  in  a straight  line  from  the  observer  ? 

12.  A ship  sailing  47°  east  of  north  changes  its  latitude  28  mi.  in 
3 hr.  What  is  its  rate  of  sailing  per  hour  ? 

13.  A ship  sailing  63°  east  of  south  changes  its  latitude  45  mi.  in 
5 hr.  What  is  its  rate  of  sailing  per  hour  ? 

14.  From  the  top  of  a lighthouse  100  ft.,  including  the  instrument, 
above  the  level  of  the  sea  a boat  is  observed  under  an  angle  of  depres- 
sion of  22°.  How  far  is  the  boat  from  the  point  of  observation  ? 

15.  Seen  from  a point  on  the  ground  the  angle  of  elevation  of  the 
top  of  a telegraph  pole  27  ft.  high  is  28°.  How  far  is  it  from  the 
point  of  observation  to  the  top  of  the  pole  ? 

16.  What  is  the  length  of  the  hypotenuse  of  a right  triangle  of 
which  one  side  is  Ilf  in.  and  the  opposite  angle  43°  ? 


FUNCTIONS  OF  ACUTE  ANGLES 


23  / 


22.  Functions  as  Lines.  The  functions  of  an  angle,  being  ratios,  are 
numbers ; but  we  may  represent  them  by  lines  if  we  first  choose  a unit 
of  length,  and  then  construct  right  tri- 
angles, such  that  the  denominators  of 
the  ratios  shall  be  equal  to  this  unit. 

Thus  in  the  annexed  figure  the 
radius  is  taken  as  1,  the  circle  then 
being  spoken  of  as  a unit  circle.  Then 
OA^OP=OB=  1. 

Drawing  the  four  perpendiculars  as 
shown,  we  have : 

MP 


siB.x  = ~ = MP-, 

tana:  = — = AT; 
OT 

sec  a;  = = 0T\ 

iJA. 


021 

GOSX  = = 02I-, 

BS 

Gotx  = — = BS-, 
OS 

CSC£C  = — — = OS. 
OB 


In  each  case  we  have  arranged  the  fraction  so  that  the  denominator  is  1. 

2IP  A T 

For  example,  instead  of  taking for  tan  x we  have  taken  the  equal  ratio , 

because  OA  = 1. 

OP  OS 

Similarly,  instead  of  taking for  esc  x we  have  taken  the  equal  ratio , 

because  OB  = 1. 

This  explains  the  use  of  the  names  tangent  and  secant,  A T being 
a tangent  to  the  circle,  and  OT  being  a secant. 

Formerly  the  functions  were  considered  as  lines  instead  of  ratios  and  received 
their  names  at  that  time.  The  word  sine  is  from  the  Latin  sinus,  a translation 
of  an  Arabic  term  for  this  function. 

We  see  from  the  figure  that  the  sine  of  the  complement  of  x 
is  NP,  which  equals  OM ; also  that  the  tangent  of  the  complement 
of  X is  BS,  and  that  the  secant  of  the  complement  of  x is  OS. 


Exercise  12.  Functions  as  Lines 

1.  Kepresent  by  lines  the  functions  of  45°. 

2.  Kepresent  by  lines  the  functions  of  an  acute  angle  greater 
than  45°. 

Using  the  ahoj)e  figure,  deterr^ne  which  is  the  greater : 

3.  sina:ortan^.  5.  see  a:  or  tan  x.  7.  cos  a;  or  cot  a. 

4.  sin  X or  sec  x.  6.  esc  x or  cot  a:.  8.  cos  x or  esc  x. 


24 


PLANE  TRIGONOMETEY 


Construct  the  angle  x,  given  the  following : 

9.  tan  a:  = 3.  11.  cosa;  = ^.  13.  sin  a:  = 2 coax. 

10.  csca:  = 2.  12.  sin  a:  = cos  a;.  14.  4 sin  a;  = tan  a:. 

16.  Show  that  the  sine  of  an  angle  is  equal  to  one  half  the  chord 

of  twice  the  angle  in  a unit  circle. 

16.  Find  x if  sin  x is  equal  to  one  half  the  side  of  a regular  deca- 
gon inscribed  in  a unit  circle. 

Given  X and  y^x  + y being  less  than  90°,  construct  a line  equal  to : 

17.  sin  (a: -h  ?/)  — sin  a:.  20.  cos  a:  — cos(a: -f  ?/). 

18.  tan  (a;  + ^)  — tan  x.  21.  cot  x — cot(a:  -f-  y~). 

19.  sec  {x  H-  ?/),—  sec  x.  22.  esc  x — esc  (x  -|-  y). 

23.  tan(a:  y)—  sin (x  y)-\-  tan x — sin x. 


Given  an  angle  x,  construct  an  angle  y such  that  : 

24.  sin  ?/ = 2 sin  ic.  28.  tan  ?/ = 3 tan  a;. 

25.  cos  cos  a;.  29.  secy  = cscx. 

26.  siny  = cosx.  30.  sin  tan  x. 

27.  tan?/ = cot X.  31.  siny  = ftanx. 


32.  Show  by  construction  that  2 sin^  > sin  2^,  when  A < 45°. 

33.  Show  by  construction  that  cos  A<  2 cos 2 A,  when  A<  30°. 

34.  Given  two  angles  A and  B,  A being  less  than  90°;  show 
that  sin  (A  + E)  < sin  A -f  sin  B. 

35.  Given  sinx  in  a unit  circle;  find  the  length  of  a line  in  a 
circle  of  radius  r corresponding  in  position  to  sinx. 

36.  In  a right  triangle,  given  the  hypotenuse  c,  and  sinA=??i; 
find  the  two  sides, 

37.  In  a right  triangle,  given  the  side  b,  and  tan  A = m;  find  the 
other  side  and  the  hypotenuse. 


Construct,  or  show  that  it  is  impossible  to  construct,  the  angle  x, 
given  the  following : 

38.  sin  X = ^.  41.  cosx  = 0.  44.  tanx  = J. 

39.  sin  X = 1.  42.  cos  x = ^.  45.  cot  x = 4. 

40.  sin  X = |.  43.  cos  x = \.  46.  sec  x = 4. 

47.  Using  a protractor,  draw  the  figure  to  show  that  sin  60°  = 
cos(-^  af  60°),  and  sin  30°  = cos  (2  x 30°). 


FUNCTIONS  OF  ACUTE  ANGLES  25 

23.  Changes  in  the  Functions.  If  we  suppose  Z.AOP,  or  x,  to  in- 
crease gradually  to  90°,  the  sine  MP  increases  to  M'P',  M”P",  and  so 
on  to  OB. 

That  is,  the  sine  increases  from  0 for  the 
angle  0°,  to  1 for  the  angle  90°.  Hence  0 and 
1 are  called  the  limiting  values  of  the  sine. 

Similarly,  AT  and  OT  gradually  in- 
crease in  length,  while  OM,  BS,  and  OS 
gradually  decrease.  That  is, 

As  an  acute  angle  increases  to  90°,  its 
sine,  tangent,  and  secant  also  increase,  while 
its  cosine,  cotangent,  and  cosecant  decrease. 

If  we  suppose  x to  decrease  to  0°,  OP  coin- 
cides with  OA  and  is  parallel  to  BS.  Therefore 
MP  and  AT  vanish,  OM  becomes  equal  to  OA,  while  BS  and  OS  are  each 
infinitely  long  and  are  represented  in  value  by  the  symbol  co.  Similarly,  we 
may  consider  the  changes  as  x increases  from  0°  to  90°. 

' Hence,  as  the  angle  x increases  from  0°  to  90°,  we  see  that 
/ I sinx  increases  from  0 to  1, 

. - cosx  decreases  from  1 to  0, 

tan  X increases  from  0 to  oo, 
cot  X decreases  from  oo  to  0, 
secx  increases  from  1 to  oo, 

CSC  X decreases  from  oo  to  1. 

' We  also  see  that 

sines  and  cosines  are  never  greater  than  1 ; 

secants  and  cosecants  are  never  less  than  1 ; 

tangents  and  cotangents  may  have  any  values  from  0 to  oo. 

In  particular,  for  the  angle  0°,  we  have  the  following  values ; 
sin  0°  = 0,  tan  0°  = 0,  sec  0°  = 1, 

cos  0°  = 1,  cot  0°  = 00,  CSC  0°  = 00. 

For  the  angle  90°  we  have  the  following  values  : 

sin  90°  = 1,  tan  90°  = oo,  sec  90°  = oo, 

cos  90°  = 0,  cot  90°  = 0,  CSC  90°  = 1. 

By  reference  to  the  figure  and  the  table  it  is  apparent  that  the  functions  of 
46°  are  never  equal  to  half  of  the  corresponding  functions  of  90°.  Thus, 
sin  46°  = 0.7071,  tan  45°  = 1,  sec  45°  = 1.4142, 

cos  46°  = 0.7071,  cot  46°  = 1,  esc  46°  = 1.4142. 


PLAi^E  TPIGONOMETEY 


H6 


, Exercise  13.  Functions  as  Lines 

/I.  Draw  a figure  to  show  that  sin  90°  = 1. 

2.  What  is  the  value  of  cos  90°  ? Draw  a figure  to  show  this. 

3.  What  is  the  value  of  sec  0°  ? Draw  a figure  to  show  this. 

4.  What  is  the  value  of  tan  90°  ? Draw  a figure  to  show  this. 

5.  What  is  the  value  of  cot  90°  ? Draw  a figure  to  show  this. 

6.  As  the  angle  increases,  which  increases  the  more  rapidly,  the 
sine  or  the  tangent  ? Show  this  by  reference  to  the  figure. 

7.  If  you  double  an  angle,  does  this  double  the  sine  ? Show  this 
by  reference  to  the  figure. 

8.  If  you  bisect  an  angle,  does  this  bisect  the  tangent  ? Prove  it. 

9.  State  the  angle  for  which  these  relations  are  true : 
sin  X = cos  X,  tan  x = cot  x,  sec  x = esc  x. 

Show  this  by  reference  to  the  figure. 


10.  If  you  know  that  sin  40°  15' = 0.6461,  and  cos  40°  15'  = 0.7632, 
and  that  the  difference  between  each  of  these  and  the  sine  and  cosine 
of  40°  15'  30"  is  0.0001,  what  is  sin  40°  15'  30"  ? cos  40°  15'  30"  ? 

11.  If  you  know  that  tan  20°  12'  is  0.3679,  and  that  the  difference 
between  this  and  tan  20°  12'  15"  is  0.0001,  what  is  tan  20°  12'  15"  ? 


12.  If  you  know  that  cot  20°  12'  is  2.7179,  and  that  the  difference 
between  this  and  cot  20°  12'  15"  is  0.0006,  what  is  cot  20°  12'  15"  ? 

13.  If  you  know  that  tan  66.5°  is  2.2998,  and  that  the  difference 
between  this  and  tan  66.6°  is  0.0111,  what  is  tan  66.6°? 

14.  If  you  know  that  cos  57.4°  is  0.5388,  and  that  the  difference 
between  this  and  cos  57.5°  is  0.0015,  what  is  cos  57.5°  ? 


Draw  the  angle  x for  which  the  functions  have  the  following  values 
and  state  (j>age  11')  to  the  nearest  degree  the  value  of  the  angle : 


16.  sin  X = 0.1. 

16.  sin  X = 0.4. 

17.  sin  X = 0.7. 

18.  cos  X = 0.9. 

19.  cos  X = 0.8. 

20.  cos  X = 0.7. 


21.  tan  a;  = 0.1. 

22.  tana;  = 0.23. 

23.  tan  x = 0.4. 

24.  cot  x = 4.0. 

26.  cot  X = 2.9. 
26.  cot  X = 0.9. 


27.  sec  a:  = 1.2. 

28.  secx  = 1.3. 

29.  seca:  = 1.7. 

30.  esex  = 2.0. 

31.  esex  = 3.6. 

32.  CSC  X = 1.66. 


33.  Pind  the  value  of  sin  x in  the  equation  sin  x ^ h 1.5  = 0. 

sinx 

Which  root  is  admissible  ? Why  is  the  other  root  impossible  ? 


CHAPTER  II 


USE  OF  THE  TABLE  OF  NATURAL  FUNCTIONS 

24.  Sexagesimal  and  Decimal  Fractions.  The  ancients,  not  having 
developed  the  idea  of  the  decimal  fraction  and  not  having  any  con- 
venient notation  for  even  the  common  fraction,  used  a system  based 
upon  sixtieths.  Thus  they  had  units,  sixtieths,  thirty-six  hun- 
dredths, and  so  on,  and  they  used  this  system  in  all  kinds  of  theo- 
retical work  requiring  extensive  fractions. 

For  example,  instead  of  1-^^  they  would  use  1 28',  meaning  l|^  ; and  instead 
of  1.51  they  would  use  1 30'  36",  meaning  l|-^  -1-  -yff-Q.  The  symbols  for  de- 
grees, minutes,  and  seconds  are  modern. 

We  to-day  apply  these  sexagesimal  (scale  of  sixty)  fractions  only 
to  the  measrrre  of  time,  angles,  and  arcs.  Thus 

3 hr.  10  min.  15  sec.  means  (3  + -|^  + -g-H-o)  hr., 
and  3°  10'  15"  means  (3  + + tIwo)° 

In  medieval  times  the  sexagesimal  system  was  carried  farther  than  this.  For 
example,  3 10'  20"  30'"  45*^  was  used  for  3 -f  — — -p —.  Some 

’ 60  602  003  ^ 604 

writers  used  sexagesimal  fractions  in  which  the  denominators  extended  to  60^2. 

Since  about  the  year  1600  we  have  had  decimal  fractions  with 
which  to  work,  and  these  have  gradually  replaced  sexagesimal  frac- 
tions in  most  cases.  At  present  there  is  a strong  tendency  towards 
using  decimal  instead  of  sexagesimal  fractions  in  angle  measure.  On 
this  account  it  is  necessary  to  be  familiar  with  tables  which  give 
the  functions  of  angles  not  only  to  degrees  and  minutes,  but  also  to 
degrees  and  hundredths,  with  provision  for  finding  the  functions  also 
to  seconds  and  to  thousandths  of  a degree.  Hence  the  tables  which 
will  be  considered  and  the  problems  which  will  be  proposed  will  in- 
volve both  sexagesimal  and  decimal  fractions,  but  with  particular 
attention  to  the  former  because  they  are  the  ones  still  commonly  used. 

The  rise  of  the  metric  system  in  the  nineteenth  century  gave  an 
impetus  to  the  movement  to  abandon  the  sexagesimal  system.  At  the 
time  the  metric  system  was  established  in  France,  trigonometric  tables 
were  prepared  on  the  decimal  plan.  It  is  only  within  recent  years, 
however,  that  tables  of  this  kind  have  begun  to  come  into  use. 

27 


28 


PLANE  TRIGONOMETEY 


25.  Sexagesimal  Table.  The  following  is  a portion  of  a page  from 
the  Wentworth-Smith  Trigonometric  Tables  : 

41°  42° 


f 

sin  cos  tan  cot 

f 

0 

6561  7547  8693  1.1504 

60 

1 

6563  7545  8698  1.1497 

59 

2 

6565  7543  8703  1.1490 

58 

3 

6567  7541  8708  1.1483 

57 

4 

6569  7539  8713  1.1477 

56 

5 

6572  7538  8718  1.1470 

55 

t 

cos  sin  cot  tan 

f 

48° 


t 

sin  cos  tan  cot 

f 

0 

6691  7431  9004  1.1106 

60 

1 

6693  7430  9009  1.1100 

59 

2 

6696  7428  9015  1.1093 

58 

3 

6698  7426  9020  1.1087 

57 

4 

6700  7424  9025  1.1080 

56 

5 

6702  7422  9030  1.1074 

55 

t 

cos  sin  cot  tan 

f 

47° 


The  functions  of  41°  and  any  number  of  minutes  are  found  by 
reading  down,  under  the  abbreviations  sin,  cos,  tan,  cot. 


For  example,  sin  41°  = 0.6561, 

cos  41°  2'  = 0.7543, 
tan  41°  4'  = 0.8713, 
cot  41°  5'  = 1.1470, 


sin  42°  = 0.6691, 

cos  42°  = 0.7431, 

tan  42°  3'  = 0.9020, 
cot  42°  5'=  1.1074. 


Decimal  points  are  usually  omitted  in  the  tables  when  it  is  obvious  where 
they  should  be  placed. 

The  secant  and  cosecant  are  seldom  given  in  tables,  being  reciprocals  of 
the  cosine  and  sine.  We  shall  presently  see  that  we  rarely  need  them. 


Since  sin  41°  2'  is  the  same  as  cos  48°  58'  (§  8),  we  may  use  the 
same  table  for  48°  and  any  number  of  minutes  by  reading  up,  above 
the  abbreviations  cos,  sin,  cot,  tan. 


For  example,  cos  48°  55'  = 0.6572, 
sin  48°  56'  = 0.7539, 
cot  48°  58'  = 0.8703, 
tan  48°  59'=  1.1497, 


cos  47°  55'  = 0.6702, 
sin  47°  56' =0.7424, 
cot  47°  57'  = 0.9020, 
tan  47°  59'=  1.1100. 


Trigonometric  tables  are  generally  arranged  with  the  degrees  from 
0"  to  44°  at  the  top,  the  minutes  being  at  the  left;  and  with  the 
degrees  from  45°  to  89°  at  the  bottom,  the  minutes  being  at  the  right. 
Therefore,  in  looking  for  functions  of  an  angle  from  0°  to  44°  59', 
look  at  the  top  of  the  page  for  the  degrees  and  in  the  left  column 
for  the  minutes,  reading  the  number  below  the  proper  abbreviation. 
For  functions  of  an  angle  from  45°  to  90°  (89°  60'),  look  at  the  bot- 
tom of  the  page  for  the  degrees  and  in  the  right-hand  column  for 
the  minutes,  reading  the  number  above  the  proper  abbreviation. 


NATUEAL  FUNCTIONS 


29 


Exercise  14.  Use  of  the  Sexagesimal  Table 

From  the  table  on  page  28  find  the  values  of  the  following : 


1.  cos  41°. 

2.  tan  42°. 

3.  cos  41°  1'. 

4.  tan  42°  2'. 

5.  cos  41°  5'. 


6.  sin  48°  59'. 

7.  sin  47°  58'. 

8.  cos  48°  59'. 

9.  cos  47°  59'. 

10.  cos  48°  57'. 


11.  sin  42°  4'. 

12.  cos  47°  56'. 

13.  tan  41°  3'. 

14.  cot  48°  57'. 

15.  tan  48°  57'. 


In  the  right  triangle  A CB,  in  which  C = 90°  : 

16.  Given  c = 27  and  A = 41°  3',  find  a. 

17.  Given  c = 48  and  A - 42°  4',  find  a. 

18.  Given  c = 61  and  A = 41°  2',  find  h. 

19.  Given  c = 12  and  A = 42°  3',  find  b. 

20.  Given  5 = 24  and  A — 41°  3',  find  a. 

21.  Given  & = 28  and  A = 42°  4',  find  a. 

22.  Given  a = 42  and  A = 41°  1',  find  b. 

23.  Given  a = 60  and  d — 42°  4',  find  b. 

24.  Given  c = 86  and  A — 48°  56',  find  a. 

25.  Given  c = 92  and  A = 48°  57',  find  a. 

26.  Given  6 = 45  and  A = 47°  55',  find  a. 

27.  Given  S = 85  and  A = 47°  59',  find  a. 

28.  Given  a.  = 86  and  A = 48°  56',  find  b. 

29.  Given  a = 98  and  A = 47°  58',  find  b. 

30.  Given  5 = 67  and  s = 100,  find  A. 

31.  A hoisting  crane  has  an  arm  30  ft.  long.  When  the  arm  makes 
an  angle  of  41°  3'  with  x,  what  is  the  length  of  v ? 
what  is  the  length  of  a:  ? 

32.  In  Ex.  31  suppose  the  arm  is  raised  until 
it  makes  an  angle  of  41°  5'  with  x,  what  are  then 
the  lengths  of  y and  x ? 

33.  From  a point  128  ft.  from  a building  the  angle  of  elevation 
of  the  top  is  observed,  by  aid  of  an  instrument  5 ft.  above  the  ground, 
to  be  42°  4'.  What  is  the  height  of  the  building  ? 

34.  From  the  top  of  a building  62  ft.  6 in.  high,  including  the 
instrument,  the  angle  of  depression  of  the  foot  of  an  electric-light  pole 
is  observed  to  be  41°  3'.  How  far  is  the  pole  from  the  building  ? 


30 


PLAITE  TRIGONOMETRY 


26.  Decimal  Table.  It  would  be  possible  to  haye  a decimal  table 
of  natural  functions  arranged  as  follows : 


0 

sin  cos  tan  cot 

O 

0.0 

0000  1.0000  0000  00 

90.0 

0.1 

0017  1.0000  0017  573.0 

89.9 

0.2 

0035  1.0000  0035  286.5 

89.8 

0.3 

0052  1.0000  0052  191.0 

89.7 

0.4 

0070  1.0000  0070  143.2 

89.6 

0.5 

0087  1.0000  0087  114.6 

89.5 

O 

cos  sin  cot  tan 

O 

O 

sin  cos  tan  cot 

e 

4.0 

0698  9976  0699  14.30 

86.0 

4.1 

0715  9974  0717  13.95 

85.9 

4.2 

0732  9973  0734  13.62 

85.8 

4.3 

0750  9972  0752  13.30 

85.7 

4.4 

0767  9971  0769  13.00 

85.6 

4.5 

0785  9969  0787  12.71 

85.5 

O 

cos  sin  cot  tan 

O 

Since,  however,  the  decimal  divisions  of  the  angle  have  not  yet  become  com- 
mon, it  is  not  necessary  to  have  a special  table  of  this  kind.  It  is  quite  con- 
venient to  use  the  ordinary  sexagesimal  table  for  this  purpose  hy  simply 
referring  to  the  Table  of  Conversion  of  sexagesimals  to  decimals  and  vice  versa. 
This  table  is  given  with  the  other  Wentworth-Smith  tables  prepared  for  use 
with  this  book.  Thus  if  we  wish  to  find  sin  27.75°,  we  see  by  the  Table  of 
Conversion  that  0.75°  = 45',  so  we  simply  look  for  sin  27°  45'. 

Tor  example,  using  either  the  above  table  or,  after  conversion  to  sexagesimals, 
the  common  table,  we  see  that : 


sin  0.4°  = 0.0070, 
cos  4.1°  = 0.9974, 
tan  0.5°  = 0.0087, 
cot  4.3°=  13.30, 


sin  85.5°=  0.9969, 
cos  85.5°  = 0.0785, 
tan  85.8°=  13.62, 
cot  85.9°  = 0.0717. 


Exercise  15.  Use  of  the  Decimal  Table 


From  the  above  table  find  the  values  of  the  following : 


1.  sin  0.5°. 

6.  sin  4.1°. 

11.  sin  85.7°. 

16.  sin  89.5°. 

2.  tan  0.4°. 

7.  cos  4.3°. 

12.  sin  85.9°. 

17.  cos  85.9°. 

3.  sin  4°. 

8.  tan  4.4°. 

13.  cos  85.6°. 

18.  tan  89. 6°. 

4.  cos  4.2°. 

9.  cot  4.5°. 

14.  tan  85.9°. 

19.  cot  89.7°. 

5.  tan  4.5°. 

10.  cot  4.2°. 

15.  cot  85.6°. 

20.  cot  85.8°. 

21.  The  hypotenuse  of  a right  triangle  is  12.7  in.,  and  one  acute 
angle  is  85.5°.  Eind  the  two  perpendicular  sides. 

22.  From  a point  on  the  top  of  a house  the  angle  of  depression  of 
the  foot  of  a tree  is  observed  to  be  4.4°.  The  house,  including  the 
instrument,  is  30  ft.  high.  How  far  is  the  tree  from  the  house  ? 

23.  A rectangle  has  a base  9.5  in.  long,  and  the  diagonal  makes  an 
angle  of  4.5°  with  the  base.  Find  the  height  of  the  rectangle  and  the 
length  of  the  diagonal. 


NATUEAL  FUNCTIONS 


31 


27.  Interpolation.  So  long  as  we  wish  to  find  the  functions  of  an 
acute  angle  expressed  in  degrees  and  minutes,  or  in  degrees  and 
tenths,  the  tables  already  explained  are  sufiicient.  But  when  the 
angle  is  expressed  in  degrees,  minutes,  and  seconds,  or  in  degrees 
and  hundredths,  we  see  that  the  tables  do  not  give  the  values  of  the 
functions  directly.  It  is  then  necessary  to  resort  to  a process  called 
interpolation. 

Briefly  expressed,  in  the  process  of  interpolation  we  assume  that 
sin  42J°  is  found  by  adding  to  sin  42°  half  the  difference  between 
sin  42°  and  sin  43°. 

In  general  it  is  evident  that  this  is  not  true.  For  example,  in 
the  annexed  figure  the  line  values  of  the  functions  of  30°  and  60° 
are  shown.  It  is  clear  that  sin  30°  is  more  than  half  sin  60°,  that 
tan  30°  is  less  than  half  tan  60°,  and  that  sec  30°  is  more  than  half 

sec  60°.  This  is  also  seen  from  the  table  on  page  11,  where 

sin  30°  = 0.5000,  tan  30°  = 0.5774,  sec  30°  = 1.1547, 

sin  60°  = 0.8660,  tan  60°  = 1.7321,  sec  60°  = 2.0000. 

For  angles  in  which  the  changes  are  very  small,  interpolation  gives 
results  which  are  correct  to  the  number  of  decimal  places  given  in 
the  table. 

For  example,  from  the  table  on  page  11  we  have 

sin  42°  = 0.6691 
sin  41°  = 0.6561 

Difference  for  1°,  or  60',  = 0.0130 

Difference  for  V = of  0.0130  = 0.0002. 

Adding  this  to  sin  41°,  we  have 

sin  41°  l'  = 0.6563, 

a result  given  in  the  table  on  page  28. 

But  if  we  wish  to  find  tan  89.6°  from  tan  89.5°  and  tan  89.7°,  we  cannot 
use  this  method  because  here  the  changes  are  very  great,  as  is  always  the  case 
with  the  tangents  and  secants  of  angles  near  90°,  and  with  the  cotangents  and 
cosecants  of  angles  near  0°.  Thus,  from  the  table  on  page  30, 

tan89.7°  = 191.0 
tan89.5°=  114.6 

Difference  for  0.2°  = 76.4 

Difference  for  0.1°  = 38.2 

Adding  this  to  tan  89.5°,  tan  89.6°  = 152.8, 
whereas  the  table  shows  the  result  to  be  143.2. 

When  cases  arise  in  which  interpolation  cannot  safely  be  used,  we  resort  to 
the  use  of  special  tables  that  give  the  required  values.  These  tables  are 
explained  later.  Interpolation  may  safely  be  used  in  all  examples  given  in 
the  early  part  of  the  work. 


32 


PLANE  TRIGONOMETEY 


28.  Interpolation  applied.  The  following  examples  will  illustrate 
the  cases  which  arise  in  practical  problems.  The  student  should 
refer  to  the  Wentworth-Smith  Trigonometric  Tables  for  the  func- 
tions used  in  the  problems. 

1.  Find  sin  22°  10'  20". 

From  the  tables,  sin  22°  11' = 0.3776 

sin  22°  10'  = 0.3773 
Difference  for  1',  or  60",  the  tabular  difference  = 0.0003 

Difference  for  20"  is  of  0.0003,  or  0.0001 

Adding  this  to  sin  22°  10',  we  have 

sin22f  10'  20"  = 0.3774 

2.  Find  cos  64°  IT  30". 

From  the  tables,  cos  64°  17' = 0.4339 

cos  64°  18'  = 0.4337 
Tabular  difference  = 0.0002 

Difference  for  30"  is  of  0.0002,  or  0.0001 

Since  the  cosine  decreases  as  the  angle  increases  we  must  subtract  0.0001 
from  cos  64°  17',  which  gives  us 

cos  64°  17'  30"  = 0.4338 

3.  Find  tan  37.54°. 

By  the  Table  of  Conversion,  0.54°  = 32'  24". 

From  the  tables,  tan  37°  33' 

tan  37°  32' 

Tabular  difference 

Difference  for  24"  is  f or  0.4,  of  0.0004 
Adding  this  to  tan  37°  32',  we  have 

tan  37.54°  = tan  37°  32'  24" 

4.  Given  sin  a:  = 0.6456,  find  x. 

Looking  in  the  tables  for  the  sine  that  is  a little  less  than  0.6456,  and  for  the 
next  larger  sine,  we  have 

0.6457  = sin  40°  13' 

0.6455  = sin  40°  12' 

0.0002  = tabular  difference 

Therefore  x lies  between  40°  12'  and  40°  13'. 

Furthermore,  0.6456  = sin  x 

0.6455  = sin  40°  12' 

0.0001  = difference 

But  0.0001  is  4 of  0.0002,  the  tabular  difference,  so  that  x is  halfway  from 
40°  12'  to  40°  13'.  Therefore  we  add  4 of  60",  or  30",  to  40°  12'. 

Hence  x = 40°  12'  30". 

We  interpolate  in  a similar  manner  when  we  use  a decimal  table. 


= 0.7687 
= 0.7683 
= 0.0004 

= 0.0002 
= 0.7685 


NATUKAL  FUNCTIONS 


33 


Exercise  16.  Use  of  the  Table 
Find  the  values  of  the  following : 


1.  sin  27°  10' 30" 

2.  sin  42°  15'  30" 

3.  sin  56°  29'  40" 

4.  sin  65°  29'  40" 

5.  cos  36°  14'  30" 

6.  cos  43°  12'  20" 

7.  cos  64°  18'  45" 

8.  tan  28°  32'  20" 

9.  tan  32°  41'  30" 

10.  tan42°  38'  30" 


11.  tan  52°  10'  45". 

12.  tan  68°  12'  45". 

13.  tan 72°  15'  50". 

14.  tan  85°  17'  45". 

15.  tan  86°  15'  50". 

16.  cot5°27'30". 

17.  cot  6°  32'  45". 

18.  cot  7°  52'  50".  ' 

19.  cot  8°  40'  10". 

20.  cot  9°  20'  10". 
Then  find  cos  x. 
Then  find  cos  x. 
Then  find  sin  x. 
Then  find  cot  x. 
Then  find  cot  x. 


21.  Given  since  = 0.6391,  find  x. 

22.  Given  since  = 0.7691,  find  x. 

23.  Given  cos  ce  = 0.3174,  find  x. 

24.  Given  tan  ce  = 2.8649,  find  ce. 

25.  Given  tan  ce  = 5.3977,  find  ce. 


First  converting  to  sexagesimals,  find  the  foUoiving : 


26.  sin  25.5°. 

27.  sin  25.55°. 

28.  sin  32.75°. 

29.  sin  41.65°. 

30.  sin  64.75°. 


31.  cos  78.52°. 

32.  tan  78.59°. 

33.  cos  81.43°. 

34.  tan  82.72°. 

35.  tan  84.68°. 


36.  cos  11.25° 

37.  cot  12.32° 

38.  cot  13.54° 

39.  cot  15.48° 

40.  cot  16.62° 


Find  the  value  of  x in  each  of  the  folloioing  equations : 


41.  sin  ce  = 0.5225. 

42.  sin  ce  0.5771. 

43.  since  = 0.6601. 

44.  since  = 0.7023. 


45.  cos  ce  = 0.7853. 

46.  cos  x = 0.7716. 

47.  cos  X = 0.9524. 

48.  cos  ce  = 0.7115. 


49.  tan  ce  = 2.6395. 

50.  tana:  = 4.7625. 

51.  tan  a:  = 4.7608. 

52.  cot  ce  = 3.7983. 


53.  If  since  = 0.6431,  what  is  the  value  of  cosce  ? 

54.  If  cos  a:  = 0.7652,  what  is  the  value  of  sin  ce  ? 

55.  If  tance  = 0.6827,  what  is  the  value  of  since  ? 

56.  If  tance  = 0.6537,  what  is  the  value  of  ce  ? of  cotcc  ? 

57.  If  cotcc  = 1.6550,  what  is  the  value  of  ce  ? of  tance  ? Verifj 

the  second  result  by  the  relation  tan  ce  = 1 /cot  x. 


34 


PLANE  TRIGONOMETRY 


29.  Application  to  the  Right  Triangle.  In  §§  15-21  we  learned 
how  to  use  the  several  functions  in  finding  various  parts  of  a right 
triangle  from  other  given  parts,  the  angles  being  in  exact  degrees. 
In  § § 25-28  we  learned  how  to  use  the  tables  when  the  angles  were  not 
necessarily  in  exact  degrees.  We  shall  now  review  both  of  these  phases 
of  the  work  in  connection  with  the  solution  of  the  right  triangle. 

In  order  to  solve  a right  triangle,  that  is,  to  find  both  of  the  acute 
angles,  the  hypotenuse,  and  both  of  the  sides,  two  independent  parts 
besides  the  right  angle  must  be  given. 

In  speaking  of  the  sides  of  a right  triangle  it  should  be  repeated  that  we  shall 
refer  only  to  sides  a and  b,  the  sides  which  include  the  right  angle,  using  the 
word  hypotenuse  to  refer  to  c.  It  will  he  found  that  there  is  no  confusion  in 
thus  referring  to  only  two  of  the  three  sides  by  the  special  name  sides. 

By  independent  parts  is  meant  parts  that  do  not  depend  one  upon  another. 
For  example,  the  two  acute  angles  are  not  independent  parts,  for  each  is  equal 
to  90°  minus  the  other. 

The  two  given  parts  may  be : 

1.  An  acute  angle  and  the  hypotenuse. 

That  is,  given  A and  c,  or  B and  c.  If  A and  c are 

given,  we  have  to  find  a and  h.  The  angle  B is  known 
from  the  relation  B = 90°  — A.  If  JB  is  given,  we  can 
find  A from  the  equation  A = 90°  — B. 

2.  An  acute  angle  and  the  opposite  side. 

That  is,  given  A and  a,  or  B and  b.  If  A and  a are  given,  we  have  to  find 

B,  b,  and  c,  and  similarly  for  the  other  case. 

3.  An  acute  angle  and  the  adjacent  side. 

That  is,  given  A and  b,  or  B and  a.  If  A and  h are  given,  we  have  to  find  B, 
a,  and  c,  and  similarly  for  the  other  case. 

4.  The  hypotenuse  and  a side. 

That  is,  given  c and  a,  or  c and  b.  If  c and  a are  given,  we  have  to  find  A,  B, 
and  &,  and  similarly  for  the  other  case. 

5.  The  two  sides. 

That  is,  given  a and  b,  to  find  A,  B,  and  c.  Using  side  to  include  hypotenuse, 
we  might  combine  the  fourth  and  fifth  of  these  cases  in  one. 

In  each  of  these  cases  we  shall  consider  right  triangles  which 
have  their  acute  angles  expressed  in  degrees  and  minutes,  in  de- 
grees, minutes,  and  seconds,  or  in  degrees  and  decimal  parts  of  a 
degree  In  this  chapter  the  angles  are  given  and  required  only  to 
the  nearest  minute. 


NATURAL  FUNCTIONS 


35 


30.  Given  an  Acute  Angle  and  the  Hypotenuse.  For  example,  given 


A = 43°  17',  c = 26,  find  £,  a,  and  h. 

1.  90° -^  = 46°  43'. 

2.  - = sind ; . a — c sin^. 

c 

a 

3.  - = cos^ ; .'.  b = c cos .4. 

c 

A b 

0 

sin^  = 0.6856 

cosA=  0.7280 

c=  26 

c=  26 

41136 

4 3680 

13  712 

14  560 

a = 17.8256 

b = 18.9280 

= 17.83 

= 18.93 

As  usual,  when  a four-place  table  is  employed,  the  result  is  given  to  four 

figures  only.  The  check  is  left  for  the  student. 

31.  Given  an  Acute  Angle  and  the  Opposite  Side.  For  example,  given 

A = 13°  58',  a = 15.2,  find  B,  b,  and  c. 

1.  90° -A  =76°  2'. 

2.  ~ = cotA;  r.  b = aootA. 
a 

B 

a . , a 

3.  - — smA ; c = . • 

c smA 

^ b 0 

a =15.2,  cotA  = 4.0207 

a = 15.2,  sin  A = 0.2414 

4.0207 

62.97  = 

15.2 

2414)152000.00 

80414 

14484 

20  1035 

7160 

40  207 

4828 

h = 61.11464 

23320 

= 61.11 

21726 

In  dividing  15.2  by  0.2414,  we  adopt  the  modem  plan  of  first  multiplying 
each  by  10,000.  Only  part  of  the  actual  division  is  shown. 

Instead  of  dividing  a by  sin^  to  find  c,  we  might  multiply  a by  esc  4,  as  on 
page  22,  except  that  tables  do  not  generally  give  the  cosecants.  It  will  be  seen 
in  Chapter  III  that,  by  the  aid  of  logarithms,  we  can  divide  by  sin  A as  readily 
as  multiply  by  esc  J.,  and  this  is  why  the  tables  omit  the  cosecant. 


36 


PLANE  TKIGONOMETEY 


32.  Given  an  Acute  Angle  and  the  Adjac^t  Side.  Eor  example,  given 
A = 27°  12',  h = 31,  find  B,  a,  and  c. 

1.  B=  90°  - 62°48'. 


2.  - = tanA;  .'.a  — btdOiA. 

3.  - = cosd;  .•.  c = -• 

c cos  A 

tanA=  0.5139 
b=  31 
5139 
15  417 
a = 15.9309 
= 15.93 


B 


b = 31,  cosd  = 0.8894 
34.85  = c 
8894)310000.00 
26682 
43180 
35576 


We  might  multiply  6 by  sec  A instead  of  dividing  by  cos  A.  The  reason  for 
not  doing  so  is  the  same  as  that  given  in  § 31  for  not  multiplying  by  cscA. 


33.  Given  the  Hypotenuse  and  a Side.  For  example,  given  a = 47, 
c = 63,  find  A,  B,  and  b. 

1.  sin  A = -• 

c 

2.  90°  - A. 

3.  b = y/c^  — 

= V(c  4-  a)  (c  — a). 

In  the  case  of  V we  can,  of  course,  square  c,  square  a,  take  the  dif- 
ference of  these  squares,  and  then  extract  the  square  root.  It  is,  however,  easier 
to  proceed  by  factoring  — a?  as  shown.  This  will  be  even  more  apparent  when 
we  come,  in  Chapter  III,  to  the  short  methods  of  computing  by  logarithms. 


B 


a = 47,  c = 63 

0.7460 
63)47.0000 
44  1 
2 90 

sin  A = 0.7460  2 52 

.-.A  = 48°  15'  380 

.-.B=41°45'  378 


c -j-  a = 110 
c — a = 16 

660 
110 

e^-a^  = 1760 
.-.  6- = 1760 
.-.  b = V1760 
= 41.95 


NATURAL  FUNCTIONS  87 


34.  Given  the  Two  Sides.  For  example,  given  a = 40,  b = 27,  find 


Of  course  c can  be  found  in  other  ways.  For  example,  after  finding  tan  A wc 
can  find  A,  and  hence  can  find  sin  A.  Then,  because  sin  A = a/c,  we  have 
c = a/sin  A.  When  the  numbers  are  small,  however,  it  is  easy  to  find  c from 
the  relation  given  above. 


a = 40,  = 27 

1 0 = 1.4815 
tan  A = 1.4815 
.•.A  = 55°  59' 
.-.  R=34°  V 


- 1600 
729 
= 2329 

. c =V2329 
= 48.26 


35.  Checks.  As  already  stated,  always  apply  some  check  to  the 
results.  For  example,  in  § 34,  we  see  at  once  that  = 1600  and  6^ 
is  less  than  30^,  or  900,  so  that  is  less  than  2500,  and  c is  less 
than  50.  Hence  the  result  as  given,  48.26,  is  probably  correct. 

We  can  also  find  B independently. 

For  since  tan  B = —, 

a 

we  see  that  tanS  = = 0.6750, 

and  therefore  that  B — 34°  1'. 


Exercise  17.  The  Right  Triangle 

Solve  the  right  triangle  A CB,  in  which  C = 90°,  given  : 


1.  a = 3,  5 = 4. 

2.  a = 7,  c = 13. 

3.  a = 5.3,  A = 12°  17'. 

4.  a =10.4,  B=  43°  18'. 

5.  c = 26,  A = 37°  42'. 

6.  c =140,  R=  24°  12'. 

7.  h =19,  c = 23. 

8.  6 = 98,  c=  135.2. 

9.  6 = 42.4,  A = 32°  14'. 


10.  6 = 200,  B=  46°  11'. 

11.  a = 95,  b = 37. 

12.  a = 6,  c = 103. 

13.  a = 3.12,  5=5°  8'. 

14.  a =17,  c =18. 

15.  c = 57,  A=  38°  29'. 

16.  a + c = 18,  5 = 12. 

17.  a + c = b = 30. 

18.  a + c = 45,  b — 30. 


38 


PLANE  TEIGONOMETEY 


Solve  the  right  triangle  ACB,  in 

19.  a = 2.5,  ^=35°  10' 30". 

20.  a = 5.7,  A = 42°  12'  30". 

21.  a=  6.4,  B=  29°  18' 30". 

22.  a =7.9,  B=  36°  20' 30". 

23.  c = 6.8,  A = 29°  42'  30". 

24.  c = 360,  A = 34°  20' 30". 

25.  h = 250,  A = 41°  10'  40". 


which  C = 90°,  given  : 

26.  a = 48,  A = 25.5°. 

27.  c = 25,  A = 24.5°. 

28.  c = 40,  A = 32.55°. 

29.  c = 80,  A = 55.51°. 

30.  c = 75,  A = 63.46°. 

31.  a = 45,  .6=  50.59°. 

32.  h = 99,A  = 68.25°. 


33.  Each  equal  side  of  an  isosceles  triangle  is  16  in.,  and  one  of 
the  equal  angles  is  24°  10'.  What  is  the  length  of  the  base  ? 

34.  Each  equal  side  of  an  isosceles  triangle  is  25  in.,  and  the  ver- 
tical angle  is  36°  40'.  What  is  the  altitude  of  the  triangle  ? 

35.  Each  equal  side  of  an  isosceles  triangle  is  25  in.,  and  one  of 
the  equal  angles  is  32°  20'  30".  What  is  the  length  of  the  base  ? 

36.  Each  equal  side  of  an  isosceles  triangle  is  60  in.,  and  the  ver- 
tical angle  is  50°  30'  30".  What  is  the  altitude  of  the  triangle  ? 

37.  Find  the  altitude  of  an  equilateral  triangle  of  which  the  side 
is  50  in.  Show  three  methods  of  finding  the  altitude. 

38.  What  is  the  side  of  an  equilateral  triangle  of 
which  the  altitude  is  52  in.  ? 

39.  In  planning  a truss  for  a bridge  it  is  necessary 
to  have  the  upright  BC  = 12  ft.,  and  the  horizontal 
A C = 8 ft.,  as  shown  in  the  figure.  What  angle  does 
AB  make  with  AC?  with  BC  ? 

40.  In  Ex.  39  what  are  the  angles  if  AB  = 12  ft.  and  AC  = 9ft.  ? 

41.  In  the  figure  of  Ex.  39,  what  is  the  length  of  .BC  if  AA  = 15  ft. 
and  a:  = 62°  10'? 

42.  Two  angles  of  a triangle  are  42°  17'  and  47°  43'  respectively, 
and  the  included  side  is  25  in.  Find  the  other  two  sides. 

43.  A tangent  AB,  drawn  from  a point  A to  a circle,  makes  an  angle 
of  51°  10'  with  a line  from  A through  the  center.  If  AB  = 10  ft.,  what 
is  the  length  of  the  radius  ? 

44.  How  far  from  the  center  of  a circle  of  radius  12  in.  will  a 
tangent  meet  a diameter  with  which  it  makes  an  angle  of  10°  20'? 

45.  Two  circles  of  radii  10  in.  and  14  in.  are  externally  tangent. 
What  angle  does  their  line  of  centers  make  with  their  common 
exterior  tangent  ? 


a 


CHAPTER  III 


LOGARITHMS 

36.  Importance  of  Logarithms.  It  has  already  been  seen  that  the 
trigonometric  functions  are,  in  general,  incommensurable  with  unity. 
Hence  they  contain  decimal  fractions  of  an  infinite  number  of  places. 
Even  if  we  express  these  fractions  only  to  four  or  five  decimal  places, 
the  labor  of  multiplying  and  dividing  by  them  is  considerable.  For 
this  reason  numerous  devices  have  appeared  for  simplifying  this 
work.  Among  these  devices  are  various  calculating  machines,  but 
none  of  these  can  easily  be  carried  about  and  they  are  too  expensive 
for  general  use.  There  is  also  the  slide  rule,  an  inexpensive  instru- 
ment for  approximate  multiplication  and  division,  hut  for  trigono- 
metric work  this  is  not  of  particular  value  because  the  tables  must  be 
at  hand  even  when  the  slide  rule  is  used.  The  most  practical  device 
for  the  purpose  was  invented  early  in  the  seventeenth  century  and 
the  credit  is  chiefly  due  to  John  Napier,  a Scotchman,  whose  tables 
appeared  in  1614.  These  tables,  afterwards  much  improved  by 
Henry  Briggs,  a contemporary  of  Napier,  are  known  as  tables  of 
logarithms,  and  by  their  use  the  operation  of  multiplication  is  re- 
duced to  that  of  addition ; that  of  division  is  reduced  to  subtraction ; 
raising  to  any  power  is  reduced  to  one  multiplication;  and  the 
extracting  of  any  root  is  reduced  to  a single  division. 

Eor  the  ordinary  purposes  of  trigonometry  the  tables  of  functions 
used  in  Chapter  II  are  fairly  satisfactory,  the  time  required  for 
most  of  the  operations  not  being  unreasonable.  But  when  a problem 
is  met  which  requires  a large  amount  of  eomputation,  the  tables  of 
natural  functions,  as  they  are  called,  to  distinguish  them  from  the 
tables  of  logarithmic  functions,  are  not  convenient. 

For  example,  we  shall  see  that  the  product  of  2.417,  3.426,  517.4,  and  91.63 
can  he  found  from  a table  by  adding  four  numbers  which  the  table  gives. 

In  the  case  of  x x we  shall  see  that  the  result  can  be  found 
62.9  5.28  9283 

from  a table  by  adding  six  numbers.  

Taking  a more  difiBcult  case,  like  that  of  J i v , we  shall  see  that  it 

\711  0.379 

is  necessary  merely  to  take  one  third  of  the  sum  of  four  numbers,  after  whidi 
the  table  gives  va  the  result. 


39 


40 


PLANE  TEIGONOMETEY 


37.  Logarithm.  The  exponent  of  the  power  to  which  a given  mim- 
ber,  called  the  base,  must  be  raised  in  order  to  be  equal  to  another 
given  number  is  called  the  logarithm  of  this  second  given  nnmhp.r. 

For  example,  since  10^  = 100, 
we  have,  to  the  base  10,  2 = the  logarithm  of  100. 

In  the  same  way,  since  10^  = 1000, 
we  have,  to  the  base  10,  3 = the  logarithm  of  1000. 

Similarly,  4 = the  logarithm  of  10,000, 

5 = the  logarithm  of  100,000, 
and  so  on,  whatever  powers  of  10  we  take. 

In  general,  if  = N, 

then,  to  the  base  &,  z = the  logarithm  of  N. 

38.  Symbolism.  For  "logarithm  of  iV”  it  is  customary  to  write 
"logiV.”  If  we  wish  to  specify  log  N to  the  base  b,  we  write  log^iV, 
reading  this  "logarithm  of  iV  to  the  base  b.” 

That  is,  as  above,  log  100  = 2,  log  10,000  = 4, 

log  1000  = 3,  log  100,000  = 5, 

and  so  on  for  the  other  powers  of  10. 

39.  Base.  Any  positive  number  except  unity  may  be  taken  as  the 
base  for  a system  of  logarithms,  but  10  is  usually  taken  for  purposes 
of  practical  calculation. 


Thus,  since 

2®  =8, 

logjS 

= 3; 

since 

31  = 81, 

logs  81 

= 4: 

and  since 

5*  = 625, 

logj  625 

= 4. 

It  is  more  convenient  to  take  10  as  the  base,  however.  For  since 
102  = 100  and  10^  = 1000, 

we  can  tell  at  once  that  the  logarithm  of  any  number  between  100  and  1000 
must  lie  between  2 and  3,  and  therefore  must  be  2 + some  fraction.  That  is, 
by  using  10  as  the  base  we  know  immediately  the  integral  part  of  the  logarithm. 

When  we  write  log  27,  we  mean  log^j27  ; that  is,  the  base  10  is  to  be  imder- 
stood  unless  some  other  base  is  specified. 

Since  log  10  - 1,  because  10^  = 10, 

and  log  1 = 0,  because  10®  =1, 

and  log^=  — 1,  because  10-^=^, 

we  see  that  the  logarithm  of  the  base  is  always  1,  the  logarithm  of  1 
is  always  zero,  and  the  logarithm  of  a proper  fraction  is  negative. 

That  this  is  true  for  any  base  is  apparent  from  the  fact  that 
61  = 6,  whence  log6  6=1; 

6®  = 1,  whence  logjl  =0; 

6-"  = —,  whence  log;,— =— n. 

6n  6» 


LOGARITHMS 


41 


13.  log.^343. 

14.  logg512. 


Exercise  18.  Logarithms 

1.  Since  2®  = 32,  what  is  log^  32  ? . 

2.  Since  4^  = 16,  what  is  log^  16  ? 

3.  Since  10^  = 10,000,  what  is  log  10,000  ? 

Write  the  following  logarithms  : 

4.  logjlO.  8.  logg243.  12.  logg36. 

5.  loggOl.  9.  loggT29. 

6.  log2l28.  10.  log^256. 

7.  log2256.  11.  loggl25.  15.  logg6561. 

20.  Since  10“^  = ^,  or  0.1,  what  is  log  0.1  ? 

21.  What  is  log  -j-^,  or  log  0.01  ? log  0.001  ? log  0.0001  ? 

22.  Between  what  consecutive  integers  is  log  52  ? log  726  ? 
log  2400?  log  24,000?  log  175,000?  log  175,000,000  ? 

23.  Between  what  consecutive  negative  integers  is  log  0.08  ? 
log  0.008  ? log  0.0008  ? log  0.1238  ? log  0.0123  ? log  0.002768  ? ^ 

24.  To  the  base  2,  write  the  logarithms  of  2,  4,  8,  64,  512,  1024, 

X ^ -J—  _L  1 1 

4>Te;  32>  64J  128J  256" 

25.  To  the  base  3,  write  the  logarithms  of  3,  81,  729,  2187,  6561, 

1 1 _L  1 1 1 1 ^ 

3)  2 7 j “ST  ’ 2437  729’  2187' 


16.  log  100. 

17.  log  1000. 

18.  log  100,000. 

19.  log  1,000,000. 


26.  To  the  base  10,  write  the  logarithms  of  1,  0.0001,  0.00001, 

10,000,000,  100,000,000. 

Write  the  consecutive  integers  between  which  the  logarithms  of 
the  following  numbers  lie : 

27.  75.  31.  642.  35.  7346.  39.  243,481. 

28.  75.9.  32.  642.75.  36.  7346.9.  40.  5,276,192. 

29.  75.05.  33.  642.005.  37.  7346.09.  ^41.  7,286,348.5. 

30.  82.95.  34.  793.175.  38.  9182.735.  42.  19,423,076. 


Show  that  the  following  statements  are  true : 

^'^3.  log24  + log28  4-  log2l6  + log^64  + log22  + logg32  = 21. 

44.  log33  + loggO  + logg81  + logg729  + logg27  + logg243  = 21, 

45.  log^ll  + log^l21  + log^,  1331  + log^  14,641  =.  10. 

46.  log  1 + log  10  + log  1000  + log  0.1  + log  0.001  = 0. 

47.  log  1 + log  100  + log  10,000  + log  0.01  + log  0.0001  = 0. 

48.  log  10,000  — log  1000  + log  100,000  — log  100  = 4. 


42 


PLANE  TEIGONOMETRY 


40.  Logarithm  of  a Product.  The  logarithm  of  the  product  of  two 
numbers  is  equal  to  the  sum  of  the  logarithms  of  the  nurribers. 

Let  A and  B be  the  numbers,  and  x and  y their  logarithms.  Then, 
taking  10  as  the  base  and  remembering  that  x = logA,  and  y = log B, 
we  have  ^ _ ^qx 

and  B — 10*'. 

Therefore  AB  = 10^  + ’^, 

and  therefore  logAB—  x + y 

= log  A 4-  logP. 

The  proof  is  the  same  if  any  other  base  is  taken.  For  example, 
if  z = logi,  A,  we  have  A = ; 

and  if  y — log;,  B,  we  have  B = bv. 

Therefore  AB  = b^  + v, 

and  log6  AB  = X + y 

= logs  A + logs  B. 

The  proposition  is  also  true  for  the  product  of  more  than  two  numbers,  the 
proof  being  evidently  the  same.  Thus, 

log  ABC  — log  A + logB  + log  C, 
and  so  on  for  any  number  of  factors. 


41.  Logarithm  of  a Quotient.  The  logarithm  of  the  quotient  of  two 
numbers  is  equal  to  the  logarithm  of  the  dividend  minus  the  logarithm 
of  the  divisor. 

For  if  A = 1(F, 

and  B = 10*', 

then  — = 10*-!', 

and  therefore  log  — = a;  — y 

--  log  A — logE. 


This  proposition  is  true  if  any  base  h is  taken.  For,  as  in  § 40, 


and  therefore 


logs 


■ X—  y 


= logs  A - logs  B. 


It  is  therefore  seen  from  §§  40  and  41  that  if  we  know  the  logarithms  of  all 
numbers  we  can  find  the  logarithm  of  a product  by  addition  and  the  logarithm 
of  a quotient  by  subtraction.  If  we  can  then  find  the  numbers  of  which  these 
results  are  the  logarithms,  we  shall  have  solved  our  problems  in  multiplication 
and  division  by  merely  adding  and  subtracting. 


LOGAEITHMS 


43 


42.  Logarithm  of  a Power.  The  logarithm  of  a power  of  a number 
is  equal  to  the  logarithm  of  the  number  multiplied  by  the  exponent. 

For  if  A = 1(F, 

raising  to  the  ^th  power, 

Hence  log^^  - - px 

= ^logJ.. 

This  is  easily  seen  by  taking  special  numbers.  Thus  if  we  take  the  base  2, 
we  have  the  following  relations  : 

Since  2®  = 32,  then  log2  32  = 6 ; 

and  since  (2®)^  = 32^  = 1024,  then  logj  1024  = 2-5 

= 2 logg  32. 

That  is,  logg  32^  = 21og2  32. 

43.  Logarithm  of  a Root.  The  logarithm  of  a root  of  a number  is 
equal  to  the  logarithm  of  the  number  divided  by  the  index  of  the  root. 

For  if  A = 10"^, 

1 X 

taking  the  rth  root,  ^.’•=10''. 


_ log^ 

r 

The  propositions  of  §§  42  and  43  are  true  whatever  base  is  taken,  as  may 
easily  be  seen  by  using  the  base  h. 

From  §§42  and  43  we  see  that  the  raising  of  a number  to  any  power,  integral 
or  fractional,  reduces  to  the  operation  of  multiplying  the  logarithm  by  the  ex- 
ponent (integral  or  fractional)  and  then  finding  the  number  of  which  the  result 
is  the  logarithm. 

Therefore  the  operations  of  multiplying,  dividing,  raising  to  powers,  and 
extracting  roots  will  be  greatly  simplified  if  we  can  find  the  logarithms  of  num- 
bers, and  this  will  next  be  considered. 

44.  Characteristic  and  Mantissa.  Usually  a logarithm  consists  of 
an  integer  plus  a decimal  fraction. 

The  integral  part  of  a logarithm  is  called  the  characteristic. 

The  decimal  part  of  a logarithm  is  called  the  mantissa. 

Thus,  if  log  2353  = 3.37162,  the  characteristic  is  3 and  the  mantissa  0.37162. 
This  means  that  l03-8n62  = 2353,  or  that  the  100,000th  root  of  the  337,162d 
power  of  10  is  2353,  approximately. 

It  must  always  be  recognized  that  the  mantissa  is  only  an  approximation, 
correct  to  as  many  decimal  places  as  are  given  in  the  table,  but  not  exact. 
Computations  made  with  logarithms  give  results  which,  in  general,  are  correct 
only  to  a certain  number  of  figures,  but  results  which  are  sufficiently  near  the 
correct  result  to  answer  the  purposes  of  the  problem. 


44 


PLANE  TRIGONOMETRY 


45.  Finding  the  Characteristic.  Since  we  know  that 

10=>  = 1000  and  10^  ==  10,000, 
therefore  3 = log  1000  and  4 = log  10,000. 

Hence  the  logarithm  of  a number  between  1000  and  10,000  lies 
between  3 and  4,  and  so  is  3 plus  a fraction.  Thus  the  characteristic 
of  the  logarithm  of  a number  between  1000  and  10,000  is  3. 

Likewise,  since 

10-3  = 0.001  and  lO-^^O.Ol, 
therefore  — 3 = log  0.001  and  — 2 = log  0.01. 

Hence  the  logarithm  of  a number  between  0.001  and  0.01  lies 
between  — 3 and  — 2,  and  so  is  — 3 plus  a fraction.  Thus  the  char- 
acteristic of  the  logarithm  of  a number  between  0.01  and  0.001  is  — 3. 

Of  course,  instead  of  saying  that  log  1475  is  3 -|-  a fraction,  we  might  say  that 
it  is  4 — a fraction ; and  instead  of  saying  that  log  0.007  is  — 3 -t-  a fraction, 
we  might  say  that  it  is  — 2 — a fraction.  For  convenience,  however,  the  man- 
tissa of  a logarithm  is  always  taken  as  positive,  but  the  characteristic  may  be 
either  positive  or  negative. 

46.  Laws  of  the  Characteristic.  From  the  reasoning  set  forth  in 
§ 45  we  deduce  the  following  laws  : 

1.  The  characteristic  of  the  logarithm  of  a number  greater  than  1 
is  positive  and  is  one  less  than  the  number  of  integral  places  in  the 
number. 

For  example,  log  75  = 1 + some  mantissa, 

log  472.8  = 2 -1-  some  mantissa, 
and  log  14,800.75  = 4-1-  some  mantissa. 

2.  The  characteristic  of  the  logarithm  of  a number  between  0 and  1 
is  negative  and  is  one  greater  than  the  number  of  zeros  between  the 
decimal  point  and  the  first  significant  figure  in  the  number. 

For  example,  log  0.02  = — 2 -1-  some  mantissa, 

and  log  0.00076  = — 4 -f  some  mantissa. 

The  logarithm  of  a negative  number  is  an  imaginary  number,  and  hence  such 
logarithms  are  not  used  in  computation. 

47.  Negative  Characteristic.  If  log  0.02  = — 2 -t-  0.30103,  we  cannot 
write  it  — 2.30103,  because  this  would  mean  that  both  mantissa  and 
characteristic  are  negative.  Hence  the  form  2.30103  has  been  chosen, 
which  means  that  only  the  characteristic  2 is  negative. 

That  is,  2.30103  =-2  -f  0.30103,  and  5.48561  = - 5 -1-  0.48561.  We  may  also 
write  2.30103  as  0.30103  — 2,  or  8.30103  — 10,  or  in  any  similar  manner  which 
will  show  that  the  characteristic  is  negative. 


LOGAKITHMS 


45 


48.  Mantissa  independent  of  Decimal  Point.  It  may  be  shown  that 
103-37107  ^ 2350  ; whence  log  2350  = 3.37107. 

Dividing  2350  by  10,  we  have 

103.87107-1  ^ 102-3713T  = 235 ; whence  log  235  = 2.37107. 

Dividing  2350  by  10^,  or  10,000,  we  have 

103.37107  - 4^  101.37107  ^ 0.235 ; whence  log  0.235  = T.37107. 

That  is,  the  mantissas  are  the  same  for  log  2350,  log  235,  log  0.235, 
and  so  on,  wherever  the  decimal  points  are  placed. 

The  mantissa  of  the  logarithm  of  a number  is  unchanged  by  any 
change  in  the  position  of  the  decimal  point  of  the  number. 

This  is  a fact  of  great  importance,  for  if  the  table  gives  us  the  mantissa  of 
log  235,  we  know  that  we  may  use  the  same  mantissa  for  log  0.00235,  log  2.35, 
log  23,500,  log  235,000,000,  and  so  on. 

Exercise  19.  Logarithms 


Write  the  characteristics  of  the  logarithms  of  the  following : 


1.  75. 

6.  2578. 

11.  0.8. 

16.  0.0007. 

2.  75.4. 

7.  257.8. 

12.  0.08. 

17.  0.0077. 

3.  754. 

8.  25.78. 

13.  0.88. 

18.  0.00007, 

4.  7.54. 

9.  2.578. 

14.  0.885. 

19.  0.10007, 

5,  7540. 

10.  25,780. 

15.  0.005. 

20.  0.07007. 

Given  3.68681  as  the  logarithm  of  3862,  find  the  following : 

21.  log  38.62.  24.  log  38,620.  27.  log  0.3862. 

22.  log  3.862.  25.  log  386,200.  28.  log  0.03862. 

23.  log  386.2.  26.  log  38,620,000.  \/29.  log  0.0003862. 

Given  1.67724  as  the  logarithm  of  0.4756,  find  the  following . 

30.  log  4756.  32.  log  47,560.  34.  log  0.04756. 

31.  log  4.756.  33.  log  47,560,000.  \y^5.  log  0.00004756, 

Given  3.40603  as  the  logarithm  of  2547,  find  the  following : 

36.  log  2.547.  38.  log  0.2547.  40.  log  25,470. 

37.  log  25.47.  39.  log  0.002547.  ’'/41.  log  25,470,000. 

Given  1.39794  as  the  logarithm  of  25,  find  the  following : 

42.  log  2^.  44.  log  0.25.  46.  log  25,000. 

43.  log:|^.  45.  log  0.025.  /47.  iQg  25,000,000, 


46 


PLANE  TKIGONOMETEY 


49.  Using  the  Table.  The  following  is  a portion  of  a page  taken 
from  the  Wentworth-Smith  Logarithmic  and  Trigonometric  Tables : 


250  — 300 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

250 

39 

794 

39 

811 

39 

829 

39 

846 

39 

863 

39 

881 

39 

898 

39 

915 

39 

933 

39 

950 

251 

39 

967 

39 

985 

40 

002 

40 

019 

40 

037 

40 

054 

40 

071 

40 

088 

40 

106 

40 

123 

252 

40 

140 

40 

157 

40 

175 

40 

192 

40 

209 

40 

226 

40 

243 

40 

261 

40 

278 

40 

295 

253 

40 

312 

40 

329 

40 

346 

40 

364 

40 

381 

40 

398 

40 

415 

40 

432 

40 

449 

40 

466 

254 

40 

483 

40 

500 

40 

518 

40 

535 

40 

552 

40 

569 

40 

586 

40 

603 

40 

620 

40 

637 

255 

40 

654 

40 

671 

40 

688 

40 

705 

40 

722 

40 

739 

40 

756 

40 

773 

40 

790 

40 

807 

Only  the  mantissas  are  given ; the  characteristics  are  always  to  be 
determined  by  the  laws  stated  in  § 46.  Always  write  the  characteristic 
at  once,  before  writing  the  mantissa. 

For  example,  looking  to  the  right  of  251  and  under  0,  and  writing  the  proper 
characteristics,  we  have 

log251  = 2.39967,  log25.1  = 1.39967, 

log  2510  = 3.39967,  log  0.0251  = 2.39967. 

The  first  three  significant  figures  of  each  number  are  given  under 
N,  and  the  fourth  figure  under  the  columns  headed  0, 1,  2, . . . , 9. 

For  example,  log  252.1  = 2.40157,  log  0.2547  = 1.40603, 

log  25.25  = 1.40226,  log  2549  = 3.40637. 

Furthermore,  log  251.1  = 2.39985  — , the  minus  sign  being  placed  beneath 
the  final  5 in  the  table  to  show  that  if  only  a four-place  mantissa  is  being  used 
it  should  be  written  3998  instead  of  3999. 

The  logarithms  of  numbers  of  more  than  four  figures  are  found  by 
interpolation,  as  explained  in  § 27. 

For  example,  to  find  log  25,314  we  have 

log  25,320  = 4.40346 
log  25,310  = 4.40329 
Tabular  difierence  = 0.00017 

.f 

0.000068 

Difference  to  be  added  = 0.00007 

Adding  this  to  4.40329,  log  25314  = 4.40336 

In  general,  the  tabular  difierence  can  be  found  so  easily  by  inspection  that 
it  is  unnecessary  to  multiply,  as  shown  in  this  example.  If  any  multiplication  is 
necessary,  it  is  an  easy  matter  to  turn  to  pages  46  and  47  of  the  tables,  where 
will  be  found  a table  of  proportional  parts.  On  page  46,  after  the  number  17  in 
the  column  of  difierences  (D),  and  under  4 (for  0.4),  is  found  6.8.  In  the  same 
way  we  can  find  any  decimal  part  of  a difierence. 


LOGAEITHMS 


47 


Exercise  20.  Using  the  Table 


Using  the  table,  find  the  logarithms  of  the  following : 


1. 

2. 

9.  3485. 

17.  0.7. 

25. 

12,340. 

2. 

20. 

10.  4462. 

18.  0.75. 

26. 

12,345. 

3. 

200. 

11.  5581. 

19.  0.756. 

27. 

12,347. 

4. 

0.002. 

12.  7007. 

20.  0.7567. 

28. 

123.47. 

5. 

2100. 

13.  5285. 

21.  0.0255. 

29. 

234.62. 

6. 

2150. 

14.  68.48. 

22.  0.0036. 

30. 

41.327. 

7. 

2156. 

15.  7.926. 

23.  0.0009. 

31. 

56.283. 

8. 

2.156. 

16.  834.8. 

24.  0.0178. 

32. 

0.41282. 

33.  In  a certain  computation  it  is  necessary  to  find  the  sum  of  the 
logarithms  of  45.6,  72.8,  and  98.4.  What  is  this  sum  ? 

34.  In  a certain  computation  it  is  necessary  to  subtract  the  loga- 
rithm of  3.84  from  the  sum  of  the  logarithms  of  52.8  and  26.5. 
What  is  the  resulting  logarithm  ? 


Perform  the  following  operations : 

35.  log  275  + log  321  + log  4.26  + log  3.87  + log  46.4. 

36.  log  2643  + log  3462  -|-  log  4926  + log  5376  -f  log  2194. 

37.  log  51.82  + log  7.263  + log  5.826  log  218.7  + log  3275. 

__  38.  log  8263  + log  2179  + log  3972  — log  2163  — log  178. 

39.  log  37.42  + log  61.73  -f-  log  5.823  — log  1.46  — log  27.83. 

40.  log  3.427  + log  38.46  + log  723.8  — log  2.73  — log  21.68. 

41.  In  a certain  operation  it  is  necessary  to  find  three  times 
log  41.75.  What  is  the  resulting  logarithm? 

42.  In  a certain  operation  it  is  necessary  to  find  one  fifth  of 
log  254.8.  What  is  the  resulting  logarithm  ? 


Perform  the  following  operations : 


43.  2 X log  3. 

44.  3 X log  2. 

45.  3 X log  25.6. 

46.  5 X log  3.76. 

47.  4 X log  21.42. 

48.  5 X log  346.8. 

49.  12  X log  42.86. 


50.  ^log2. 

51.  ^ log  2000. 

52.  -J- log  3460. 

53.  ^ log  24.76. 

54.  log  368.7. 

55.  flog 41.73. 
66.  flog 763.8. 


^57.  0.3  log  431. 

58.  0.7  log  43.19. 

59.  0.9  log  4.007. 

60.  1.4  log  5.108. 

61.  2.3  log  7.411. 

62.  I log  16.05. 

63.  flog 23.43. 


48 


PLANE  TRIGONOMETRY 


50.  Antilogarithm.  The  number  corresponding  to  a given  logarithm 
is  called  an  antilogarithm. 

For  " antilogarithm  of  N ” it  is  customary  to  -write  " antilog  N.” 

Thus  if  log  25.31  = 1.40329,  antilog  1.40329  = 25.31.  Similarly,  -we  see  that 
antilog  5.40329  = 253,100,  and  antilog  2.40329  = 0.02531. 

51.  Finding  the  Antilogarithm.  An  antilogarithm  is  found  from 
the  tables  by  looking  for  the  number  corresponding  to  the  given 
mantissa  and  placing  the  decimal  point  according  to  the  character- 
istic. For  example,  consider  the  following  portion  of  a table : 


550  — 600 


N 

0 

1 

2 

3 4 

6 

6 7 8 9 

660 

551 

74  036 
74  115 

74  044 
74123 

74  052  74  060  74  068 
74  131  74  139  74  147 

74  076 
74  155 

74  084  74  092  74  099  74  107 
74  162  74  170  74  178  74  186 

If  the  mantissa  is  given  in  the  table,  we  find  the  sequence  of  the 
digits  of  the  antilogarithm  in  the  column  under  N.  If  the  mantissa 
is  not  given  in  the  table,  we  interpolate. 


1.  Find  the  antilogarithm  of  5.74139. 

We  find  74139  in  the  table,  opposite  551  and  under  3.  Hence  the  digits  of  the 
number  are  5513.  Since  the  characteristic  is  5,  there  are  six  integral  places, 
and  hence  the  antilogarithm  is  551,300.  That  is, 
log  551,300  = 5.74139, 
or  antilog  5.74139  = 551,300. 

2.  Find  the  antilogarithm  of  2.74166. 

We  find  74170  in  the  table,  opposite  551  and  under  7. 

log  0.05517  = 2.74170 
logO.05516  = 2.74162. 

Tabular  difference  = 0.00008 

Subtracting,  -we  see  that,  neglecting  the  decimal  point,  the  tabular  difference 
is  8,  and  the  difference  between  log  x and  log  0.05516  is  4.  Hence  x is  | of  the 
way  from  0.05516  to  0.05517.  Hence  x = 0.055165. 

3.  Find  the  antilogarithm  of  7.74053. 

We  find  74060  in  the  table,  opposite  550  and  under  3. 

log  55,030,000  = 7.74060 
log  55,020,000  = 7.74052 
Tabular  difference  = 0.00008 

Seasoning  as  before,  x is  ^ of  the  way  from  55,020,000  to  55,030,000. 
Hence,  to  five  significant  figures,  x = 55,021,000. 

In  general,  the  interpolation  gives  only  one  additional  figure  correct ; that  is, 
with  a table  like  the  one  above,  the  sixth  figure  will  not  be  correct  if  found  by 
interpolation. 


LOGARITHMS 


4^ 

Exercise  21.  Antilogarithms 


Find  the  antilogarithms  of  the  following : 


1.  0.47712. 

9.  3.74076. 

17.  0.23305. 

25. 

8.77425. 

2.  3.47712. 

10.  2.76305. 

18.  1.43144. 

26. 

4.82966. 

/3.  3.47712. 

11.  4.78497. 

19.  2.56838. 

27. 

3.83547. 

4.  2.48359. 

12.  T. 81954. 

20.  1.58041. 

28. 

2.83604. 

5.  4.56844. 

13.  0.82575. 

21.  3.63490. 

29. 

4.88960. 

6.  1.66276. 

14.  0.88081. 

22.  4.63492. 

30. 

2.89523. 

7.  2.66978. 

15.  9.89237. 

23.  0.63994. 

31. 

3.89858. 

8.  5.74819. 

16.  7.90282. 

24.  2.69085. 

/32.  0.93223. 

^ 33.  If  the  logarithm  of  the  product  of  two  numbers  is  2.94210, 
what  is  the  product  of  the  numbers  ? 

^ 34.  If  the  logarithm  of  the  quotient  of  two  numbers  is  0.30103, 
what  is  the  quotient  of  the  numbers  ? 

35.  If  we  wish  to  multiply  2857  by  2875,  what  logarithms  do  we 
need  ? What  are  these  logarithms  ? 

36.  If  we  know  that  the  logarithm  of  a result  which  we  are  seek- 
ing is  3.47056,  what  is  that  result? 

^ 37.  If  we  know  that  log  V0.000043641  is  3.81995,  what  is  the 
value  of  Vo.000043641  ? 

38.  If  we  know  that  log  '^0.076553  is  1.81400,  what  is  the  value 
of  -^0.076553  ? 

39.  The  logarithm  of  V8322  is  1.96012.  Find  V8322  to  three 
decimal  places. 

40.  The  logarithm  of  the  cube  of  376  is  7.72557.  Find  the  cube 
of  376  to  five  significant  figures. 

41.  If  we  know  that  log  0.003278^  is  5.03122,  what  is  the  value  ^ 
of  0.003278"  ? 

42.  Find  twice  log  731,  and  find  the  antilogarithm  of  the  result. 

43.  Find  the  antilogarithm  of  the  sum  of  log  27.8  + log  34.6 + 
log  367.8. 

Find  the  antilogarithms  of  the  following : 

44.  log  7 -j-  log  2 — log  1.934.  47.  5 log  27.83. 

45.  log  63  -j-  log  5.8  — log  3.415.  48.  2.8  log  5.683. 

46.  log  728  -1-  log  96.8  — log  2.768.  \)  49.  f (log  2 -|-  log  4.2). 


60 


PLANE  TRIGONOMETEY 


52.  Multiplication  by  Logarithms.  It  has  been  shown  (§  40)  that 
the  logarithm  of  a product  is  equal  to  the  sum  of  the  logarithms  of 
the  numbers.  This  is  of  practical  value  in  multiplication. 

Find  the  product  of  6.15  x 27.05. 

From  the  tables,  log  6.15  = 0.78888 

log  27.06  = 1.43217 
log  a;  =2.22105 

Interpolating  to  find  the  value  of  x,  we  have 

log  166.4  = 2.22116  logx  =2.22105 

log  166.3  = 2.22089  log  166.3  = 2.22089 

26  16 
Annexing  to  166.3  the  fraction  , we  have 

X = 166.3|| 

= 166.36, 

the  interpolation  not  being  exact  beyond  one  figure. 

If  we  perform  the  actual  multiplication,  we  have  6.15  x 27.05  = 166.3676,  or 
166.36  to  two  decimal  places. 

Exercise  22.  Multiplication  by  Logarithms 


Using  logarithms,  find  the  following  products : 


1. 

2 

X 

5. 

11. 

2 X 50. 

21. 

35.8  X 28.9. 

2. 

4 

X 

6. 

12. 

40 

X 60. 

22. 

52.7  X 41.6. 

3. 

3 

X 

5. 

13. 

3 X 500. 

23. 

2.75  X 4.84. 

4. 

5 

X 

7. 

14. 

50 

X 70. 

24. 

5.25  X 3.86. 

6. 

2 

X 

4. 

15. 

2 X 4000. 

25. 

14.26  X 42.35. 

6. 

3 

X 

7. 

16. 

30 

X 700. 

26. 

43.28  X 29.64. 

7. 

2 

X 

6. 

17. 

200  X 60. 

27. 

529.6  X 348.7. 

8. 

3 

X 

6. 

18. 

30 

X 600. 

28. 

240.8  X 46.09. 

9. 

7 

X 

8. 

19. 

7 >< 

c 80,000. 

29. 

34.81  X 46.25. 

10. 

2 

X 

9. 

20. 

200  X 900. 

30. 

5028  X 3.472. 

31.  Taking  the  circumference  of  a circle  to  be  3.14  times  the 
diameter,  find  the  circumference  of  a steel  shaft  of  diameter  5.8  in. 

32.  Taking  the  ratio  of  the  circumference  to  the  diameter  as  given 
in  Ex.  31,  find  the  circumference  of  a water  tank  of  diameter  36  ft. 

Using  logarithms,  find  the  following  products : 

33.  2 X 3 X 5 X 7.  36.  43.8  X 26.9  x 32.8. 

34.  3 X 5 X 7 X 9.  37.  527.6  x 283.4  x 4.196. 

36.  5 X 7 X 11  X 13.  38.  7.283  X 6.987  x 5.437. 


LOGAEITHMS 


51 


53.  Negative  Characteristic.  Since  the  mantissa  is  always  positive 
(§  45),  care  has  to  be  taken  in  adding  or  subtracting  logarithms  in 
which  a negative  characteristic  may  occur.  In  all  such  cases  it  is 
better  to  separate  the  characteristics  from  the  mantissas,  as  shown 
in  the  following  illustrations  : 

1.  Add  the  logarithms  2.81764  and  1.41283. 

Separating  the  negative  characteristic  from  its  mantissa,  we  have 

2.81764  = 0.81764  - 2 
1.41283  = 1.41283 

Adding,  we  have  2.23047  — 2 

= 0.23047 

2.  Add  the  logarithms  4.21255  and  2.96245. 

Separating  both  negative  characteristics  from  the  mantissas,  we  have 
4.21255  = 0.21255  - 4 
2.96245  = 0.96245  - 2 

Adding,  we  have  1. 17500  ^ 6 

= 5.17500 


Exercise  23.  Negative  Characteristics 


Add  the  following  logarithms : 

1.  2.41283  + 5.27681. 

2.  2.41283  + 5.27681. 

3.  2.41283  + 5.27681. 

4.  0.38264  + 4.71233. 

5.  0.57121  + 1.42879. 


6.  2.63841  + 1.36158. 

7.  2.41238  + 3.62701. 

8.  5.58623  + 6.41387. 

9.  6.41382  + 7.58617. 
flO.  4.22334  + 3.77666. 


Using  logarithms,  find  the  following  products : 


11.  256  X 4875. 

12.  2.56  X 48.75. 

13.  0.256  X 0.4875. 

14.  0.0256  X 0.004875. 

15.  0.1275  X 0.03428. 

16.  0.2763  X 0.4134. 

17.  0.00025  X 0.00125. 

25. 

26. 

27. 

V 


18.  0.725  X 0.3465. 

19.  0.256  X 0.0875. 

20.  0.037  X 0.00425. 

21.  47.26  X 0.02755. 

22.  296.8  X 0.1283 

23.  45,650  X 0.0725. 

^ 24.  127,400  X 0.00355. 


Given  sin  25.75°  = 0.4344,  find  52.8  sin  25.75°. 

Given  cos  37.25°  = 0.7960,  find  42.85  cos  37.25°. 

Given  tan  30°  50' 30"  = 0.5971,  find  27.65  tan  30°  50' 30". 


52 


PLANE  TPIGONOMETEY 


54.  Division  by  Logarithms.  It  has  been  shown  (§  41) 
logarithm  of  a quotient  is  equal  to  the  logarithm  of  the 
minus  the  logarithm  of  the  divisor. 

Care  must  be  taken  that  the  mantissa  in  subtraction 
become  negative  (§  45). 

1.  Using  logarithms,  divide  17.28  by  1.44. 

Erom  the  tables,  logl7. 28  = 1.23754 

log  1.44  = 0.15836 
1.07918 
= log  12 

Hence  17.28  1.44  = 12. 

2.  Using  logarithms,  divide  2603.5  by  0.015998. 

log  2603.5  = 3.41556 

log  0.015998  = 2.20407 

Arranging  these  in  a form  more  convenient  for  subtracting,  we  have 
log  2603.5  = 3.415-56 

log  0.015998  = 0.20407  - 2 
3.21149  + 2 

= 5.21149  = log  162,740 

Hence  2603.6  --  0.016998  = 162,740. 

3.  Using  logarithms,  divide  0.016502  by  127.41. 

log  0.016502  = 2.21753  = 8.21753  - 10 
log  127.41  = 2.10520  = 2.10520 

6.11233-  10 

= 4.11233  = log  0.00012952 
Hence  0.016502  ---  127.41  = 0.00012952. 

Here  we  increased  2.21753  by  10  and  decreased  the  sum  by  10.  We  might 
take  any  other  number  that  would  make  the  highest  order  of  the  minuend 
larger  than  the  corresponding  order  of  the  subtrahend,  but  it  is  a convenient 
custom  to  take  10  or  the  smallest  multiple  of  10  that  will  serve  the  purpose. 

4.  Using  logarithms,  divide  0.000148  by  0.022922. 

log  0.000148  = 1.17026  = 16.17026  - 20 
log  0.022922  = 2.36025  = 8.36025  - 10 
7.81001  - 10 

= 3.81001  = log  0.0064567 
Hence  0.000148  h-  0.022922  = 0.0064567. 

5.  Using  logarithms,  divide  0.2548  by  0.05513. 

log  0.2548  = 1.40620  = 9. 40620  - 10 
log  0.05613  = 174139  = 8.74139-  10 
0.66481 
= log  4.6218 

Hence  0.2548  -f-  0.05513  = 4.6218. 


that  the 
dividend 

does  not 


LOGARITHMS 


53 


Exercise  24.  Division  by  Logarithms 

Add  the  following  logarithms : 

1.  2.14755  + 3.82764. 

2.  4.07256  + 1.58822. 

3.  0.21783  + 1.46835. 

4.  0.41722  + 3.28682. 


5.  4.18755  + 2.81245. 

6.  6.28742  + 3.41258. 

7.  4.21722  + 4.78278. 

8.  5.28720  + 3.71280. 


9.  rind  the  sum  of  2.41280,  4.17623,  5.26453,  0.21020,  7.36423, 
2.63577,  6.41323,  and  3.28740. 


From  the  first  of  these  logarithms  subtract  the  second  : 


10.  0.21250,  2.21250. 

11.  0.17286,3.27286. 

12.  2.34222,  A44222. 

13.  3.14725,1.25625. 


14.  4.17325,  2.17325. 

15.  5.82340,  3.71120. 

16.  3.14286,1.14000. 
^ 17.  3.27283,  5.56111. 


Using  logarithms. 
18.  10  2. 

19.  15^3. 

20.  15^5. 

21.  12 -3. 

22.  12  ^ 4. 

23.  60  ^ 12. 

24.  75  --25. 

25.  125  ^25. 


divide  as  folloivs : 

26.  25,284-- 301. 

27.  51,742^631. 

28.  47,348  ^ 623. 

29.  19,224  H-  540. 

30.  37,960  ^520. 

31.  84,640  H-  920. 

32.  65,100  ^ 620. 

33.  45,990  H- 730. 


34.  59.29  H- 0.77. 

35.  2.451 -- 190. 

36.  851.4  H-  0.66. 

37.  0.98902  --  99. 

38.  0.41831 -- 5.9. 

39.  0.08772  ^ 4.3. 

40.  0.02275  -j- 0.35. 
Al.  0.02736  --  0.057 


Using  logarithms,  divide  to  four  significant  figures : 

42.  15-^7.  45.  26.4-- 13.8.  48.  17.625  ^-3.4. 

43.  7-J-15.  46.  4.21 -^- 3.75.  49.  43.826  ^0.72. 

44.  0.7 -T- 150.  47.  63.25  ^4.92.  50.  5.483^8.4. 


Taking  log  3.1416  as  0.49715  and  interpolating  for  six  figures 
on  the  same  principle  as  for  five,  find  the  diameters  of  circles  with 
circumferences  as  follows : 

51.  62.832.  53.  2199.12.  55.  28,274.2.  57.  376,992. 

52.  157.08.  54.  2513.28.  56.  34,557.6.  58.  0.031416. 

59.  By  using  logarithms  find  the  product  of  41.74  x 20.87,  and 
the  quotient  of  41.74  --  20.87. 


54 


PLANE  TRIGONOMETRY 


55.  Cologarithm.  The  logarithm  of  the  reciprocal  of  a number  is 
called  the  cologarithm  of  the  number. 

For  "cologarithm  of  N”  it  is  customary  to  write  "colog  A.” 

By  definition  colog  x = log  - = log  1 — logo;  (§  41).  But  log  1 = 0. 

Hence  we  have  colog  x——  log  x. 

To  avoid  a negative  mantissa  (§  45)  it  is  customary  to  consider  that 
colog  a:  = 10  — log  x — 10, 
since  10  — log  a:  — 10  is  the  same  as  — log  a:. 

For  example,  colog  2 = — log  2 = 10  — log  2 — 10 

= 10  - 0.30103  - 10 
= 9.69897  - 10  = 1.69897. 

56.  Use  of  the  Cologarithm.  Since  to  divide  by  a number  is  the  same 
as  to  multiply  by  its  reciprocal,  instead  of  subtracting  the  logarithm 
of  a divisor  we  may  add  its  cologarithm. 

The  cologarithm  of  a number  is  easily  written  by  looking  at  the  logarithm 
in  the  table.  Thus,  since  log  20  = 1.30103,  we  find  colog  20  by  subtracting  this 
from  10.00000  — 10.  To  do  this  we  begin  at  the  left  and  subtract  the  number 
represented  by  each  figure  from  9,  except  the  right-hand  significant  figure, 
which  we  subtract  from  10.  In  full  form  we  have 

10.00000  - 10  = 9.  9 9 9 9 10  - 10 

log  20  = 1.30103  = 1.  3 0 1 0 3 

colog  20  = 8.  6 9 8 9 7 - 10  = 2.69897 

Similarly,  we  may  find  colog  0.03952  thus : 

10.00000  - 10  = 9.  9 9 9 9 10  - 10 

log  0.03952  = 159682  = 8.  5968  2 - 10 

cologO.03952  = 1.  4 0 3 1 8 = 1.40318 

Practically,  of  course,  we  would  find  log  0.03952  and  subtract  mentally. 


Exercise  25.  Cologarithms 


Write  the  cologarithms  of  the  following  numbers: 


1.  25. 

6.  3751. 

9.  0.5. 

13.  3.007. 

2.  130. 

6.  427.3. 

10.  0.72. 

14.  62.09. 

3.  27.4. 

7.  51.61. 

11.  0.083. 

15.  0.0006. 

4.  5.83. 

8.  7.213. 

12.  0.00726. 

16.  O.OOOOT 

17.  What  number  has  0 for  its  cologarithm  ? 

18.  What  number  has  1 for  its  cologarithm  ? 

19.  What  number  has  oo  for  its  cologarithm  ? 

20.  Find  the  number  whose  cologarithm  equals  its  logarithm. 


LOGARITHMS 


55 


57.  Advantages  of  the  Cologarithm.  If,  as  is  not  infrequently  the 
case  in  the  computations  of  trigonometry  and  physics,  we  have  the 
product  of  two  or  more  numbers  to  be  divided  by  the  product  of 
two  or  more  different  numbers,  the  cologarithm  is  of  great  advantage. 
Using  logarithms  and  cologarithms,  simplify  the  expression 

17.28  X 6.25  X 16.9 
1.44  X 0.25  X 1.3 

This  is  so  chosen  that  we  can  easily  verify  the  answer  by  cancellation. 

By  logarithms  we  have, 

log  17.28  = 1.23754 
log  6.25  = 0.79588 
log  16.9  = 1.22789 
colog  1.44  = 9.84164  — 10 
colog  0.25  = 0.60206 
colog  1.3  = 9.88606  - 10 

3.59107  = log  3900.1 

In  a long  computation  the  fifth  figure  may  be  in  error. 


Exercise  26.  Use  of  Cologarithms 


Using  cologarithms,  find  the  value  of  the  following  to  five  figures  •. 


3x2 

10. 

172.8  X 1.44 

19. 

435  X 0.2751 

4 xl.5 

0.288  X 0.864 

2.83  X 1.045 

8x9 

11. 

57.5  X 0.64 

20. 

50.05  X 2.742 

3x4 

1.25  X 320 

381.4  X 2.461 

6 xl2 

12. 

1.28  X 13.41 

21. 

50730  X 2.875 

3x8 

1.49  X 6.4 

34.48  X 1.462 

4 X 24 

13. 

5.48  X 0.198 

22. 

3.427  X 0.7832 

12  xl6 

3.96  X 27.4 

3.1416  X 0.0081 

12  xl5 

14. 

1.176  X 10.22 

23. 

27.98  X 32.05 

9 X 20 

14.6  X 3.92 

0.48  X 0.00062 

12  X 28 

15. 

3 X 11  X 17 

24. 

2.1  X 0.3  X 0.11 

8 X 21 

7 xl3 

17  X 0.05 

3 X 22 

16. 

16  X 23 

V 26. 

1.1  X 3.003 

18  X 33 

3 X 7 X 41 

0.2  X 0.07112 

8. 


11x13 

17x19’ 


23  X 39  X 47 
17  X 33  X 53* 


. 0.0347  X 0.117 

• 3 X 11  X 170 


15  xl7  0.2  X 0.3 

11x13'  0.11  xl7i' 


528.4  X 1.001 
7.03  X 0.7281' 


9. 


27. 


56 


PLA^E  TRIGONOMETRY 


58.  Raising  to  a Power.  It  has  been  shown  (§  42)  that  the  logarithm 
of  a power  of  a number  is  equal  to  the  logarithm  of  the  number 
multiplied  by  the  exponent. 

1.  Find  by  logarithms  the  value  of  11®. 

From  the  tables,  log  11  =1.04139 

Multiplying  by  3,  3 

log  118  ^ 3.12417 
= log  1331.0 

That  is,  11®  = 1331.0,  to  five  figures.  Of  course  we  see  that  11®  = 1331  exactly, 
log  1331  being  3.12418.  The  last  figure  in  log  11®  as  found  in  the  above  multh 
plication  is  therefore  not  exact,  as  is  frequently  the  case  in  such  computations. 

As  usual,  care  must  be  taken  when  a negative  characteristic 
appears. 

2.  Find  by  logarithms  the  value  of  0.2413®. 

From  the  tables,  log  0.2413  = 0.38256  — 1 

Multiplying  by  6,  6 

log  0.2413®  = 1.91280-6 
- 4.91280 
= log  0.00081808 

Hence  0.2413®  = 0.00081808,  to  five  significant  figures. 

As  on  page  18,  we  use  the  expression  "significant  figures”  to  indicate  the 
figures  after  the  zeros  at  the  left,  even  though  some  of  these  figures  are  zero. 

Exercise  27.  Raising  to  Powers 

By  logarithms,  find  the  value  of  each  of  the  following  to  five 
significant  figures: 


1. 

2®. 

9. 

1“. 

17.  25®. 

25. 

1.1®. 

33. 

12.55®. 

2. 

2®. 

10. 

7®. 

18.  25h 

26. 

2.V. 

34. 

34.75®. 

3. 

2®. 

11. 

9’. 

19.  125®. 

27. 

0.11®. 

35. 

1.275®. 

4. 

2“ 

12. 

8®. 

20.  625®. 

28. 

0.211. 

36. 

0.1254®. 

6. 

3®. 

13. 

IV. 

21.  1750®. 

29. 

0.7®. 

37. 

0.4725®. 

6. 

3®. 

14. 

15®. 

22.  2775®. 

30. 

o 

o 

-0 

38. 

0.01234®. 

7. 

4®. 

15. 

1.5®. 

23.  3146®. 

31. 

0.37h 

^ 39. 

0.00275®. 

8. 

5®. 

16. 

17h 

24.  41351 

32. 

5.37®. 

V40. 

0.000355®. 

41.  If  log  7T  = 0.49716,  what  is  the  value  of  tt®  ? of  tt®  ? 

42.  U sing  log  TT  as  in  Ex.  41,  what  is  the  value  of  wr  when  r = 7 ? 
of  XT®  when  ?*  = 7 ? of  f Trr®  when  ?•  = 9 ? 


LOGARITHMS 


57 


59.  Fractional  Exponent.  It  has  been  shown  (§  43)  that  the  log- 
arithm of  a root  of  a number  is  equal  to  the  logarithm  of  the  number 
divided  by  the  index  of  the  root.  This  law  may,  however,  be  com- 
bined with  that  of  § 58,  since  means  Va,  and  means 
The  law  of  § 58  therefore  applies  to  roots  or  to  powers  of  roots,  the 
exponent  simply  being  considered  fractional. 

1.  Find  by  logarithms  the  value  of  Vl,  or  4^. 

From  the  tables,  log  4 = 0.60206 

Dividing  by  2,  2)0.60206 

log  Vi,  or  log  4^,  = 0.30103 
= log2 

Hence  Vi,  or  4^,  is  2. 

2.  Find  by  logarithms  the  value  of  8^. 

From  the  tables,  log  8 = 0.90309 

Multiplying  by  logSt  = 0.60206 

= log  4 

Therefore  8^  = 4. 

3.  Find  by  logarithms  the  value  of  0.127^. 

From  the  tables,  log  0.127  = 0.10380  — 1. 

Since  we  cannot  divide  — 1 by  6 and  get  an  integral  quotient  for  the  new 
characteristic,  we  add  4 and  subtract  4 and  then  have 

log  0.127  = 4.10380-5 

Dividing  by  5,  log  0. 127^  = 0.82076  — 1 

= log  0.66185 

Hence  0.127^,  or  Vo.l27,  is  0.66185. 

We  might  have  written  log  0.127  = 9.10380  — 10,  14.10380  — 15,  and  so  on. 

Exercise  28.  Extracting  Roots 


By  logarithms,  find  the  value  of  each  of  the  follovnng : 


1. 

V2. 

5.  2L 

9.  Vil. 

13.  0.3^ 

iai.  127.8i 

2. 

^5. 

6.  3L 

10.  73. 

14.  0.05L 

18.  2.475i 

3. 

77. 

7.  8i 

11.  7^. 

15.  0.0175^. 

19.  5.135i 

4. 

7^. 

8.  7L 

12.  TiM. 

16.  0.0325^ 

V20.  0.00125f 

21.  If  log  7T  = 0.49715,  what  is  the  value  of  Vtt  ? of  ? 

22.  Using  the  value  of  log  tt  given  in  Ex.  21,  what  is  the  value  of 
7T^  ? of  7T^  ? of  ? of  7T“i  ? of  7r“^  ? of  ? 


58 


PLANE  TRIGONOMETRY 


60.  Exponential  Equation.  An  equation  in  which  the  unknown 
quantity  appears  in  an  exponent  is  called  an  exponential  equation. 

Exponential  equations  may  often  be  solved  by  the  aid  of  loga- 
rithms. 


1.  Given  5"^  = 625,  find  by  logarithms  the  value  of  x. 


Taking  the  logarithms  of  both  sides,  we  have  (§  42) 
X log  5 =:  log  625 
_ log  625 


Whence 


log  5 


Check.  5*  = 625. 


2.79588 

0.69897 


In  all  such  cases  bear  in  mind  that  one  logarithm  must  actually  be  divided 
by  the  other.  If  we  wished  to  perform  this  division  by  means  of  logarithms, 
we  should  have  to  take  the  logarithm  of  2.79588  and  the  logarithm  of  0.69897, 
subtract  the  second  logarithm  from  the  first,  and  then  find  the  antilogarithm. 

We  may  apply  this  principle  to  certain  simultaneous  equations. 


2.  Solve  this  pair  of  simultaneous  equations 

2*  • 3*'  = 72  (1) 

4*  . 27>'  = 46,656  (2) 

Taking  the  logarithms  of  both  sides,  we  have  (§§  40,  42) 

X log 2 -f- 2/ logs  = log  72,  (3) 

and  X log  4 + y log  27  = log  46,656.  (4) 

Then,  since  log  4 = log2^  = 2 log  2, 

and  log  27  = log  3*  = 3 log  3, 

we  have  2 x log  2 4- 3 y log  3 = log  46,656.  (5) 


Eliminating  x by  multiplying  equation  (3)  by  2 and  subtracting  from  equa- 
tion (5),  we  have 

_ log  46656  — 2 log  72 

i^^3 

_ 4.66890-  2 X 1.85733 
“ 0.47712 

_ 0.95424  _ ^ 

“ 0.47712  “ 

We  may  substitute  this  value  of  y in  (1),  divide  by  3*,  and  then  find  x by 
taking  the  logarithms  of  both  sides.  It  will  be  found  that  x = 3. 

We  may  check  by  substituting  in  (2). 

In  the  same  way,  equations  involving  three  or  more  unknown 
quantities  may  be  solved.  Although  the  exponential  equation  is 
valuable  in  algebra,  as  in  the  solution  of  Exs.  22,  23,  25,  and  26  of 
Exercise  29.  we  rarely  have  need  of  it  in  trigonometry. 


LOGARITHMS 


69 


Exercise  29.  Exponential  Equations 

By  logarithms,  solve  the  following  exponential  equations: 


V 1.  2^  = 8. 

2.  3*  = 81. 

3.  5"  = 625. 

4.  4"  - 256. 

6.  1H  = 1331. 


6.  2"^  =19. 

7.  3"^  ==75. 

8.  5"  = 1000. 

9.  4"  = 2560. 
10.  11^=1500. 


'■Al.  2-^  = f 
12.  2-^  = 0.1. 

13.  0.3- * = 0.9. 

14.  2*+^  = 3^-\ 
v'lS.  9"^+®  = 53,143. 


Solve  the  following  simultaneous  equations : 


^ 16.  + = a^ 

a^~'>  — c? 

17.  ^ mS- 


18.  3"^  . 4*'  =12 
5"  . 7*'  = 35 

19.  2*  • 3*'  = 36 
4^  . 5^  = 400 


20.  2"^ . 5"  = 200 
3^  . 3^  = 243 

21.  2^  . 8*'  = 256 

8^  . 32*'  = 65,536 


Solve  the  following  equations  by  logarithms : 

- 22.  a=p(l+rf.  ^25.  a—p(l-\-rty. 

V 23.  l=ar^~^.  26.  s(r —I)  — ar^  — a. 

24.  2^+^^  — 8.  ^'27.  3=^- "'+1  = 27. 


Perform  the  following  operations  by  logarithms : 


„ 2.47  X 84.96 

34.8  X 96.55' 


/ 5.75  X 3.428 
V59.62  X 48.08/  ’ 


29. 


i 


42.4  X 0.075 
3.64  X 0.009' 


J 


31 


■ NV3. 


07  X 0.00964Y 
426  X 0.875  / 


32.  To  what  power  must  7 be  raised  to  equal  117,649  ? 

33.  To  wbat  power  must  a be  raised  to  equal  h ? 

34.  To  wbat  power  must  5 be  raised  to  equal  n ? 

35.  Find  tbe  value  of  x when  "v^  = 3 ; when  ^f2  = 1.1 ; when 

^ = 1.414 ; when  = 1.73. 

36.  Find  tbe  value  of  x when  "V^  = 3 ; when  "Va  = h ; when 
Va  = a j when  •^1331  = 11 ; when  ■v'^20736  = 12. 

\5  37.  Solve  tbe  equations 

Vi/  = a 

i+i/-  , 

Sy  = h 

1 

I 38.  Wbat  value  of  x satisfies  tbe  equation  = Va? 


60 


PLANE  TPIGONOMETPY 


61.  Logarithms  of  the  Functions.  Since  computations  involving 
trigonometric  functions  are  often  laborious,  they  are  generally  per- 
formed by  the  aid  of  logarithms.  Eor  this  reason  tables  have  been 
prepared  giving  the  logarithms  of  the  sine,  cosine,  tangent,  and 
cotangent  of  the  various  angles  from  0°  to  90°  at  intervals  of  1'. 
The  functions  of  angles  greater  than  90°  are  defined  and  discussed 
later  in  this  work  when  the  need  for  them  arises. 

Logarithms  of  the  secant  and  cosecant  are  usually  not  given  for  the  reason 
that  the  secant  is  the  reciprocal  of  the  cosine,  and  the  cosecant  is  the  reciprocal 
of  the  sine.  Instead  of  multiplying  by  secx,  for  example,  we  may  divide  by 
cos  X ; and  when  we  are  using  logarithms  one  operation  is  as  simple  as  the  other, 
since  multiplication  requires  the  addition  of  a logarithm  and  division  requires 
the  addition  of  a cologarithm. 

In  order  to  avoid  negative  characteristics  the  characteristic  of 
every  logarithm  of  a trigonometric  function  is  printed  10  too  large, 
and  hence  10  must  be  subtracted  from  it. 

Practically  this  gives  rise  to  no  confusion,  for  we  can  always  tell  by  a result 
if  a logarithm  is  10  too  large,  since  it  would  give  an  antilogarithm  with  10 
integral  places  more  than  it  should  have.  For  example,  if  we  are  measuring 
the  length  of  a lake  in  miles,  and  find  10.30103  as  the  logarithm  of  the  result, 
we  see  that  the  characteristic  must  be  much  too  large,  since  this  would  make 
the  lake  20,000,000,000  mi.  long. 

It  would  be  possible  to  print  2.97496  for  log  sin  5°  25',  instead  of  8.97496, 
which  is  10  too  large.  It  would  be  more  troublesome,  however,  for  the  eye  to 
detect  the  negative  sign  than  it  would  be  to  think  of  the  characteristic  as 
10  too  large. 

On  pages  56-77  of  the  tables  the  characteristic  remains  the  same  throughout 
each  column,  and  is  therefore  printed  only  at  the  top  and  bottom,  except  in 
the  case  of  pages  58  and  77.  Here  the  characteristic  changes  one  unit  at  the 
places  marked  with  the  bars.  By  a little  practice,  such  as  is  afforded  on  pages 
61  and  62  of  the  text,  the  use  of  the  tables  will  become  clear. 

On  account  of  the  rapid  change  of  the  sine  and  tangent  for  very 
small  angles  log  sin  x is  given  for  every  second  from  0"  to  3'  on 
page  49  of  the  tables,  and  log  tan  x has  identically  the  same  values 
to  five  decimal  places.  The  same  table,  read  uj)wards,  gives  the 
log  cos  X for  every  second  from  89°  57'  to  90°.  Also  log  sin  cr, 
log  tan  X,  and  log  cos  x are  given,  on  pages  50-55  of  the  tables,  for 
every  10"  from  0"  to  2°.  Reading  from  the  foot  of  the  page,  the 
cofunctions  of  the  complementary  angles  are  given. 

On  pages  56-77  of  the  tables,  log  sinx,  log  cos  a-,  log  tanrr,  and 
log  cot  X are  given  for  every  minute  from  1°  to  89°.  Interpolation 
in  the  usual  manner  (page  31)  gives  the  logarithmic  functions  for 
every  second  from  1°  to  89°. 


LOGAEITHMS 


61 


62.  Use  of  the  Tables.  The  tables  are  used  in  much  the  same  way 


as  the  tables  of  natural  functions. 

For  example,  log  sin  5°  25'  = 8.97496  — 10  Page  58 

logtan40°55'  = 9.93789-10  Page  75 

log  cos  52°  20'  = 9.78609  — 10  Page  74 

log  cot  88°  59'  = 8.24910  - 10  Page  56 

logsin  0°  28'  40"  = 7.92110  - 10  Page  51 

logsin  0°  1' 52"  = 6.73479  — 10  Page  49 

Furthermore,  if  log  cot  x = 9.55910  — 10,  then  x = 70°  5'.  Page  65 


Interpolation  is  performed  in  the  usual  manner,  whether  the  angles 
are  expressed  in  the  sexagesimal  system  or  decimally. 

1.  Find  log  sin  19°  50'  30". 

From  the  tables,  logsin  19°  50' = 9.53056 —10,  and  the  tabular  difference 
is  36.  We  must  therefore  add  -|ff  of  36  to  the  mantissa,  in  the  proper  place. 
We  therefore  add  0.00018,  and  have  logsin  19°  50'  30"  = 9.53074  — 10. 

2.  Find  log  tan  39.75°. 

From  the  tables,  log  tan  39.7°  = 9.91919  — 10,  and  the  tabular  difference  is 
154.  We  therefore  add  0.5  of  154  to  the  mantissa,  in  the  proper  place.  Adding 
0.00077,  we  have  log  tan  39.75°  = 9.91996  — 10. 

Special  directions  in  the  case  of  very  small  angles  are  given  on 
page  49  of  the  tables.  It  should  be  understood,  however,  that  we 
rarely  use  angles  involving  seconds  except  in  astronomy. 

If  the  function  is  decreasing,  care  must  be  taken  to  subtract  instead 
of  add  in  making  an  interpolation. 

3.  Find  log  cos  43°  45'  15". 

From  the  tables,  log  cos  43°  45'  = 9.85876  — 10,  and  the  tabular  difference  is 
12.  Taking  ^ of  12,  or  l of  12,  we  have  0.00003  to  be  subtracted. 

Therefore  log  cos  4.3°  45'  15"  = 9.85873  — 10. 

4.  Given  log  cotx  = 0.19268,  find  x. 

From  the  tables,  log  cot  32°  41'  = 10.19275  — 10  = 0.19275. 

The  tabular  difference  is  28,  and  the  difference  between  the  logarithm  0.19275 
and  the  given  logarithm  is  7,  in  each  case  hundred-thousandths.  Hence  there  is 
an  angular  difference  of  of  1',  or  ^ of  1',  or  15".  Since  the  angle  increases  as 
the  cotangent  decreases,  and  0.19268  is  less  than  10.19275  — 10,  we  have  to 
add  15"  to  32°  41',  whence  x = 32°  41'  15". 

5.  Given  log  tanx  = 0.26629,  find  x. 

From  the  tables,  log  tan  61°  33'  = 10.26614  — 10  = 0.26614. 

The  tabular  difference  is  30,  and  the  difference  between  the  logarithm 
0.26614  and  the  given  logarithm  is  15,  in  each  case  hundred-thousandths. 
Hence  there  is  an  angular  difference  of  of  1',  or  30".  Slnce/(x)  is  increasing  in 
this  case,  and  x is  also  increasing,  we  add  30"  to  61°  S3'.  Hence  x = 61°  3.3' JO", 


62 


PLANE  TRIGONOMETRY 


Exercise  30-  Use  of  the  Tables 


Find  the  value  of 

each 

of  the  following  : 

1. 

log  sin  27°. 

16. 

log  cos  42°  45". 

31. 

log 

sin  0°  1 

'7'( 

f 

2. 

log  sin  69°. 

17. 

log  tan 26°  15". 

32. 

log 

sin  1°  2 

'5", 

3. 

log  cos  36°. 

18. 

log  cot  38°  30". 

33. 

log 

tan  0°  2 

'8" 

4. 

log  cos  48°. 

19. 

log  sin  21°  10' 4". 

34. 

log 

tan  2°  i 

"7' 

1 

6. 

log  tan  75°. 

20. 

log  sin  68°  49'  56". 

36. 

log 

cos  89° 

50' 

10" 

6. 

log  tan  12°. 

21. 

log  cos  15°  17' 3". 

36. 

log 

cos  89° 

10' 

45". 

7. 

log  cot  15°. 

22. 

log  cos  74°  42' 57". 

37. 

log 

cot  89° 

15' 

12" 

8 

log  cot  78°. 

23. 

log  tan  17°  2'  10". 

38. 

log 

cot  89° 

25' 

15" 

9. 

log  sin  9°  15'. 

24. 

log  tan  26°  3'  4". 

39. 

log 

sin  1°  1 

'1" 

10. 

log  cos  8°  27'. 

26. 

log  cot  48°  4' 5". 

40. 

log 

cos  88° 

58' 

59". 

11. 

log  tan  7°  56'. 

26. 

log  cot  4°  10'  7". 

41. 

log  tan  2°  2 

17' 2 

!5". 

12. 

log  cot  82°  4'. 

27. 

log  sin  34°  30". 

42. 

log 

cot  87° 

32' 

45". 

13. 

log  sin  4.5°. 

28. 

log  sin  27.45°. 

43.. 

log 

sin  12° 

12' 

12". 

14. 

log  cos  7.25°. 

29. 

log  tan  56.35°. 

44. 

log 

cos  77° 

47' 

48" 

16. 

log  tan  9.75°. 

\/30. 

log  cos  48.26°. 

45. 

log 

tan  68° 

6'  43". 

Find  the  value  of  x,  given  the  following  logarithms,  each  of  which 
is  10  too  large : 


46.  log  sin  a:  = 9.11570. 

69. 

log  sin  X — 9.53871. 

47.  log  sin  cc  = 9.72843. 

60. 

log  sin  a:  — 9.72868. 

48.  log  sin  X = 9.93053. 

61. 

log  sin  X = 9.88150. 

49.  log  sin  X = 9.99866. 

62. 

log  sin  a:  = 9.89530. 

60.  log  cos  a:  = 9.99866. 

63. 

log  cos  X = 9.90151. 

61.  log  cos  a:  = 9.93053. 

64. 

log  cos  a:  = 9.80070. 

62.  log  cos  a:  = 9.71705. 

65. 

log  cos  X = 9.99483. 

63.  log  cos  a:  = 9.80320. 

66. 

log  tana:  = 9.18854. 

64.  log  tana;  = 9.90889. 

67. 

log  tan  X = 10.18750. 

66.  log  tana:  = 10.30587. 

68. 

logtanx  = 10.06725. 

66.  log  tana:  = 10.64011. 

69. 

log  cot  X = 10.10134. 

67.  log  cot  a:  = 9.28865. 

v^O. 

log  cot  X = 11.44442. 

68.  log  cot  X = 9.56107. 

71. 

log  cot  X = 7.49849. 

CHAPTER  IV 


THE  RIGHT  TRIANGLE 


63.  Given  an  Acute  Angle  and  the  Hypotenuse.  In  § 30  the  solution 

of  the  right  triangle  was  considered  when  an  acute  angle  and  the 
hypotenuse  are  given.  We  may  now  consider  this  case  and  the  follow- 
ing cases  with  the  aid  of  logarithms.  For  example, 
given  A = 34°  28',  c — 18.75,  find  B,  a,  and  h. 

1.  .B  = 90°  - d = 55°  32'. 


2.  - = sin  A 
c 


.-.a  — c sin  A. 


3.  - = cosd  ; .‘.b  = c cos  A. 
c 

log  a = logc  + log  sin  A 

log  c = 1.27300 
log  sin  A = 9.7 527 6—10 
loga  = 1.02576 

.-.  a = 10.611 
- 10.61 


log  b = log  c -fi  log  cos  A 

logc  = 1.27300 
log  cos  A = 9.91617  — 10 
logJ  = 1.18917 
.-.b  = 15.459 
= 15.46 


Check.  10.612  + 15.452  = 351.58,  and  18.752  351.55. 

This  solution  may  be  compared  with  the  one  on  page  35.  In  this  case  there 
is  a gain  in  using  logarithms,  since  we  avoid  two  multiplications  by  18.75. 

The  result  is  given  to  four  figures  (two  decimal  places)  only,  the  length  of  c 
having  been  given  to  four  figures  (two  decimal  places)  only,  and  this  probably 
being  all  that  is  desired.  In  general,  the  result  cannot  be  more  nearly  accurate 
than  data  derived  from  measurement. 

Consider  also  the  case  in  which  A = 72°  27' 42",  c =147.35,  to 
find  B,  a,  and  b as  above. 


log  a = log  c -f  log  sin  A 

log  c = 2.16835 
log  sin  A = 9.97933 


■10 


log  a = 2.14768 
.'.  a = 140.50 


log  b = logc  + log  cos  A 
log  c = 2.16835 
log  cosA=  9.47906  —10 
log  b = 1.64741 

.-.  b = 44.403 


Check.  What  convenient  check  can  be  applied  in  this  case  ? 

63 


64 


PLANE  TRIGONOMETRY 


64.  Given  an  Acute  Angle  and  the  Opposite  Side.  Eor  example,  given 
A = 62°  10',  a = 7S,  find  B,  h,  and  c. 


log  b — log  a + log  cotd 
log  a = 1.89209 
log  eotd  = 9.72262  - 10 
log  b = 1.61471 
.-.  b = 41.182 
= 41.18 


log  c = log  a -f-  colog  sin  A 
log  a = 1.89209 
colog  sin  A = 0.05340 
log  c = 1.94549 
.-.  c = 88.204 
= 88.20 


GUeck.  88.202  _ 41182  _ 6083  + , and  782  = 0O84. 

This  solution  should  be  compared  with  the  one  given  in  § 31,  page  3.5.  It  will 
be  seen  that  this  is  much  shorter,  especially  as  to  that  part  in  which  c is  found. 
The  difference  is  still  more  marked  if  we  remember  that  only  part  of  the  long 
division  is  given  in  § 31. 


65.  Given  an  Acute  Angle  and  the  Adjacent  Side.  For  example, 
given  A = 50°  2',  b — 88,  find  B,  a,  and  c. 


log  a - log  b 4-  log  tan  A 
log  b = 1.94448 
log  tan  A = 10.07670  — 10 
loga=  2.02118 
.-.  a = 105.00 


log  c = log  b 4-  colog  cos  A 
log  5 = 1.94448 
colog  cos  A = 0.19223 
log  c = 2.13671 
.-.c  = 137.00 


Check.  1372  - 1062  = 7744^  and  882  = 7744. 

This  solution  should  he  compared  with  the  one  given  in  § 32,  page  36.  Here 
again  it  will  be  seen  that  a noticeable  gain  is  made  by  using  logarithms,  partic- 
ularly in  finding  the  value  of  c 


THE  EIGHT  TEIAHGLE 


65 


66.  Given  the  Hypotenuse  and  a Side.  Eor  example,  given  a — 47.55, 


We  could,  of  course,  find  b from  the  equation  6 = V(c  + a)  (c  — a),  as  in 
§ 33,  page  36.  By  taking  b = a cot^,  however,  we  save  the  trouble  of  first  find- 


ing c -1-  a and  c — a. 

log  sin  A = log  a -I-  colog  c 
log  a - 1.67715 
colog  c = 8.23359  —10 
log  sin^  = 9.91074  —10 
.-.  A=  54°  31' 

.-.  35°  29' 


log  d — log  a -|-  log  cot  A 
log  a = 1.67715 
log  cot.4  = 9.85300  —10 
log  h = 1.53015 
.-.  h = 33.896 
= 33.90 


Check.  58.42  _33.g2  = 2261-1-,  and  47.552  = 2261-1- . 

This  solution  should  be  compared  with  the  one  given  in  § 33,  page  36. 

67.  Given  the  Two  Sides.  Eor  example,  given  a = 40,  b = 27,  find 


B 


A,  B,  and  c. 

, , a 

1.  tanG  = -• 

b 

2.  90° -A. 
a 

3.  - = sin  A : 
c 

.\a  — c sinG,  and  c = . ^ • 
sinA 

log  tan  A = log  a -fi  colog  b 
log(t=  1.60206 
colog  5=  8.56864—10 
log  tan  A = 10.17070  —10 
.-.  A=  55°  59' 

.-.  B=  34°  1' 


log  c = log  a 4-  colog  sin  A 
log  a = 1.60206 
colog  sin -4  = 0.08151 
logc  - 1.68357 
.-.  c = 48.258 
= 48.26 


Check.  272  + 402  = 2329,  and  48.262  = 2329  + . 

This  solution  should  be  compared  with  the  solution  of  the  same  problem  given 
in  § 34,  page  37.  There  is  not  much  gained  in  this  particular  example  because 
the  numbers  are  so  small  that  the  operations  are  easily  performed. 


66 


PLANE  TRIGONOMETRY 


68.  Area  of  a Right  Triangle.  The  area  of  a triangle  is  equal  to  one 
half  the  product  of  the  base  by  the  altitude ; therefore,  if  a and  h 
denote  the  two  sides  of  a right  triangle  and  5'  the  area,  then  S=\ab. 

Hence  the  area  may  be  found  when  a and  b are  known. 

Consider  first  the  case  in  which  an  acute  angle  and  the  hypotenuse 
are  given.  For  example,  let  A = 34°  28'  and  c = 18.75.  Then,  finding 
log  a and  log  6 as  in  § 63,  we  have 

log  S = colog  2 + log  a + log  h 
colog  2 = 9.69897  - 10 
log  a = 1.02576 
log^  = 1.18917 
log  S = 1.91390 
.-.  S = 82.016 
= 82.02 

Next  consider  the  case  in  which  the  hypotenuse  and  a side  are 
given.  For  example,  let  c = 58.4  and  a = 47.55.  Then,  finding  log  b 
as  in  § 66,  we  have 

log  S = colog  2 + log  a + log  b 
colog  2 = 9.69897  — 10 
log  a = 1.67715 
log^>  = 1.53015 
log  S = 2.90627 
.-.  S = 805.88 
= 805.9 

Finally,  consider  the  case  in  which  an  acute  angle  and  the  opposite 
side  are  given.  For  example,  let  A =62°  10'  and  a = 78.  Then, 
finding  log  i as  in  § 64,  we  have 

log  5 = colog  2 + log  a + log  b 
colog  2 = 9.69897-10 
log  a = 1.89209 
log6  = 1.61471 
log  S = 3.20577 

.-.5  = 1606.1 
= 1606 

We  can  easily  verify  this  result,  since,  from  § 64,  a = 78  and  6 = 41.18  ; 
whence  | o6  = 1606,  to  four  significant  figures. 

The  case  of  an  acute  angle  and  the  opposite  side  is  treated  in  § 64 ; that  of 
an  acute  angle  and  the  adjacent  side  in  § 66  ; and  that  of  the  two  sides  in  § 67. 


THE  RIGHT  TRIANGLE 


67 


Exercise  31.  The  Right  Triangle 

Using  logarithms,  solve  the  following  right  triangles,  finding  the 
sides  and  areas  to  four  figures,  and  the  angles  to  minutes  : 


1. 

a = 6, 

c = 

12. 

16. 

II 

So 

CO 

o 

CO 

II 

2. 

J = 4, 

A = 

60°. 

17. 

a = 992, 

B=  76°  19' 

3. 

a = 3, 

A = 

30°. 

18. 

a = 73, 

B=  68°  52' 

4. 

a = 4, 

1 

4. 

19. 

a = 2.189, 

B=  45°  25' 

6. 

a = 2, 

r 

2.89. 

20. 

5 = 4, 

A = 37°  56' 

6. 

c = 627, 

i - : 

23°  30'. 

21. 

c - 8590, 

a = 4476. 

7. 

c = 2280, 

A = 

28°  5'. 

22. 

c = 86.53, 

a = 71.78. 

8. 

C : 72.15, 

A = 

39°  34'. 

23. 

c = 9.35, 

a = 8.49. 

9. 

c = 1, 

A = 

36°. 

24. 

c = 2194, 

5 = 1312.7. 

10. 

c = 200, 

B = 

21°  47'. 

25. 

c - 30.69, 

5 = 18.25. 

11. 

c = 93.4, 

B = 

76°  25'. 

26. 

a = 38.31, 

5 = 19.62. 

12. 

a = 637, 

A = 

4°  35'. 

27. 

a - 1.229, 

5 = 14.95. 

13. 

a = 48.63, 

A = 

36°  44'. 

28. 

a - 415.3, 

5 = 62.08. 

14. 

a = 0.008, 

A = 

86°. 

29. 

a = 13.69, 

5 = 16.92. 

15. 

b = 50.94, 

B = 

43°  48'. 

30. 

c = 91.92, 

= 2.19. 

Compute  the  unknoum  parts  and 

also  the  area,  having  given : 

31. 

a = 5, 

b = 

6. 

36. 

c = 68, 

A = 69°  54' 

32. 

a = 0.615, 

c = 

70. 

37. 

c = 27, 

B=  44°  4'. 

33. 

b=V2, 

c = 

V3. 

38. 

a = 47, 

B=  48°  49' 

34. 

a = l, 

A = 

18°  14'. 

39. 

5 = 9, 

B=  34°  44' 

35. 

5 = 12, 

A = 

29°  8'. 

40. 

c = 8.462, 

B=  86°  4'. 

41.  Find  the  value  of  S in  terms  of  c and  A. 

42.  Find  the  value  of  S in  terms  of  a and  A. 

43.  Find  the  value  of  S in  terms  of  b and  A. 

44.  Find  the  value  of  S in  terms  of  a and  c. 

46.  Given  S = 58  and  a = 10,  solve  the  right  triangle. 

46.  Given  S = 18  and  b = 5,  solve  the  right  triangle. 

47.  Given  5 = 12  and  A = 29°,  solve  the  right  triangle. 

48.  Given  S = 98  and  c = 22,  solve  the  right  triangle. 

49.  Find  the  two  acute  angles  of  a right  triangle  if  the  hypote- 
nuse is  equal  to  three  times  one  of  the  sides. 


68 


PLANE  TEIGONOMETEY 


50.  The  latitude  of  Washington  is  38°  55'  15"  N.  Taking  the 
radius  of  the  earth  as  4000  mi.,  what  is  the  radius  of  the  circle 
of  latitude  of  Washington  ? What  is  the  circum- 
ference of  this  circle  ? 

In  all  such  examples  the  earth  will  he  considered  as 
a perfect  sphere  with  the  radius  as  above  given,  unless 
the  contrary  is  stated.  For  more  accurate  data  consult 
the  Table  of  Constants. 

61.  What  is  the  difference  between  the  length  of  a degree  of  lati- 
tude and  the  length  of  a degree  of  longitude  at  Washington  ? 

Use  the  data  given  in  Ex.  50. 

52.  From  the  top  of  a mountain  1 mi.  high,  overlooking  the  sea, 
an  observer  looks  toward  the  horizon.  What  is  the  angle  of  depres- 
sion of  the  line  of  sight  ? 

In  the  figure  the  height  of  the  mountain  is  necessarily 
exaggerated.  The  angle  is  so  small  that  the  result  can  he 
found  by  five-place  tables  only  between  two  limits  which 
differ  by  3'  40". 

^ 53.  At  a horizontal  distance  of  120  ft.  from  the 

foot  of  a steeple,  the  angle  of  elevation  of  the  top  is  found  to  be 
60°  30'.  Find  the  height  of  the  steeple  above  the  instrument. 

64.  From  the  top  of  a rock  which  rises  vertically  326  ft.  out  of 
the  water,  the  angle  of  depression  of  a boat  is  found  to  be  24°. 
Find  the  distance  of  the  boat  from  the  base  of  the  rock. 

55.  How  far  from  the  eye  is  a monument  on  a level  plain  if  the 
height  of  the  monument  is  200  ft.  and  the  angle  of  elevation  of 
the  top  is  3°  30'  ? 

56.  A distance  AB  of  96  ft.  is  measured  along  the  bank  of  a river 
from  a point  A opposite  a tree  C on  the  other  bank.  The  angle  ABC 
is  21°  14'.  Find  the  breadth  of  the  river. 

57.  What  is  the  angle  of  elevation  of  an  inclined  plane  if  it  rises 
1 ft.  in  a horizontal  distance  of  40  ft.  ? 

58.  Find  the  angle  of  elevation  of  the  sun  when  a tower  120  ft. 
high  casts  a horizontal  shadow  70  ft.  long. 

59.  How  high  is  a tree  which  casts  a horizontal  shadow  SO  ft.  in 
length  when  the  angle  of  elevation  of  the  sun  is  50°  ? 

SO.  A rectangle  7.5  in.  long  has  a diagonal  8.2  in.  long.  What 
angle  does  the  diagonal  make  with  the  base  ? 


THE  EIGHT  TRIANGLE 


69 


61.  A rectangle  85  in.  long  has  an  area  of  49 J sq.  in.  Find  the 
angle  which  the  diagonal  makes  with  the  base. 

62.  The  length  AB  of  a rectangular  field  ABCD  is  80  rd.  and  the 
width  is  60  rd.  The  field  is  divided  into  two  equal  parts  by  a 
straight  fence  PQ  starting  from  a point  P on  AB  which  is  15  rd. 
from  A.  What  angle  does  PQ  make  with  AB? 

63.  A ship  is  sailing  due  northeast  at  the  rate  of  10  mi.  an  hour. 
Find  the  rate  at  which  she  is  moving  due  north,  and  also  due  east. 

64.  If  the  foot  of  a ladder  22  ft.  long  is  11  ft.  from 
a house,  how  far  up  the  side  of  the  house  does  the  lad- 
der reach? 

65.  In  front  of  a window  20  ft.  from  the  ground  there 
is  a flower  bed  6 ft.  wide  and  close  to  the  house.  How 
long  is  a ladder  which  will  just  reach  from  the  outside 
edge  of  the  bed  to  the  window  ? 

O''  66.  A ladder  40  ft.  long  can  be  so  placed  that  it  will  reach  a win- 
dow 33  ft.  above  the  ground  on  one  side  of  the  street,  and  by  tipj^ing 
it  back  without  moving  its  foot  it  will  reach  a window  21  ft.  above 
the  ground  on  the  other  side.  Find  the  width  of  the  street. 

67.  From  the  top  of  a hill  the  angles  of  depression  of  two  suc- 
cessive milestones,  on  a straight,  level  road  leading  to  the  hill, 
are  5°  and  15°.  Find  the  height  of  the  hill. 

68.  A stick  8 ft.  long  makes  an  angle  of  45°  with 
the  floor  of  a room,  the  other  end  resting  against  the 
wall.  How  far  is  the  foot  of  the  stick  from  the  wall  ? 

69.  A building  stands  on  a horizontal  plain. 

The  angle  of  elevation  at  a certain  point  on  the 
plain  is  30°,  and  at  a point  100  ft.  nearer  the 
building  it  is  45°.  How  high  is  the  building  ? 

^ 70.  From  a certain  point  on  the  ground  the  angles  of  elevation 
of  the  top  of  the  belfry  of  a church  and  of  the  top  of  the  steeple 
are  found  to  be  40°  and  51°  respectively.  From  a point  300  ft.  fur- 
ther off,  on  a horizontal  line,  the  angle  of  elevation  of  the  top  of 
the  steeple  is  found  to  be  33°  45'.  Find  the  height  of  the  top  of  the 
steeple  above  the  top  of  the  belfry. 


B 


^ 71.  The  angle  of  elevation  of  the  top  C of  an  inaccessible  fort 

^ observed  from  a point  A is  12°.  At  a point  B,  219  ft.  from  A and 
on  a line  AB  perpendicular  to  AC,  the  angle  is  61°  45'.  Find 
the  height  of  the  fort. 


70 


PLANE  TRIGONOMETPvY 


69.  The  Isosceles  Triangle.  Since  an  isosceles  triangle  is  divided 
by  tlie  perpendicular  from  the  vertex  to  the  base  into  two  congruent 
right  triangles,  an  isosceles  triangle  is  determined  by  any  two  parts 
which  determine  one  of  these  right  triangles. 

In  the  examples  which  follow  we  shall  represent  the  parts  of  the 
isosceles  triangle  ABC,  among  which  the  altitude  CD  is  included, 
as  follows : 


a = one  of  the  equal  sides, 
c = the  base, 
h = the  altitude, 

A = one  of  the  equal  angles, 
C = the  angle  at  the  vertex. 


For  example,  given  a and  c,  find  A,  C, 
and  h. 


1.  cos^ 


i£  — ^ . 

a 2 a 


2.  C + 2A  = 180°;  .-.  C = 180°  -2A  = 2(90°  - A). 

3.  h may  be  found  by  any  one  of  the  following  equations  : 

A- 1 = d-, 

whence  /;.  = V(a  + c)  (a  — ^ c) ; 

or  - = sinA,  whence  A = usinA; 

or  — = tan  A,  whence  Ti  = \ ctanA. 

When  c and  h are  known,  the  area  can  be  found  by  the  formula 
S = ^ch 

That  is,  S = • a sin  A = ^ ac  sin  A, 

or  S = I c • tan  A = J tan  A, 

or  S = ^cV(a  + i-c)  (a  — ^c). 

Consider  also  the  case  in  which  a and  /i  are  given,  to  find  A,  (7, 
c,  and  S. 

1.  sin  A = and  hence  A is  known. 

a 


2.  C = 2(90°  — A),  as  above,  and. hence  C is  known. 

3.  ^ c = a.  cos  A,  and  hence  c is  known. 

4.  iS  = -^  ch,  and  hence  S is  known. 

We  can  also  find  S from  any  of  its  other  equivalents,  such  as  those  given 
above,  or  sin  ^ G sinA,  each  of  which  is  easily  deduced. 


THE  EIGHT  TEIAIIGLE 


71 


Exercise  32.  The  Isosceles  Triangle 

Solve  the  following  isosceles  triangles : 

1.  Given  a and  A,  find  C,  c,  and  h. 

2.  Given  a and  C,  find  A,  c,  and  h. 

3.  Given  c and^,  find  C,  a,  and  A. 

4.  Given  c and  C,  find  A,  a,  and  h. 

6.  Given  h and  A,  find  C,  a,  and  c. 

6.  Given  h and  C,  find  A,  a,  and  c. 

7.  Given  a and  h,  find  A,  C,  and  c. 

8.  Given  c and  h,  find  A,  C,  and  a. 

9.  Given  a = 14.3,  c = 11,  find  A,  C,  and  h. 

10.  Given  a = 0.295,  A = 68°  10',  find  c,  h,  and  S. 

11.  Given  c = 2.352,  C = 69°  49',  find  a,  h,  and  S. 

12.  Given  h — 7.4847,  ^ = 76°  14',  find  a,  c,  and  S. 

13.  Given  c = 147,  S — 2572.5,  find  A,  C,  a,  and  h. 

14.  Given  h = 16.8,  S = 43.68,  find  A,  C,  a,  and  c. 

15.  Given  a = 27.56,  ^ = 75°  14',  find  c,  h,  and  S. 

\ 

Given  an  isosceles  triangle,  AB  C : 

16.  Find  the  value  of  S in  terms  of  a and  C. 

17.  Find  the  value  of  S in  terms  of  a and  A. 

18.  Find  the  value  of  S in  terms  of  h and  C. 

19.  A barn  is  40  ft.  by  80  ft.,  the  pitch  of  the  roof  is  45°;  find 
the  length  of  the  rafters  and  the  area  of  the  whole  roof. 

20.  In  a unit  circle  what  is  the  length  of  the  chord  subtending 
the  angle  45°  at  the  center  ? 

21.  The  radius  of  a circle  is  30  in.,  and  the  length  of  a chord  is 
44  in. ; find  the  angle  subtended  at  the  center. 

22.  Find  the  radius  of  a circle  if  a chord  whose  length  is  5 in, 
subtends  at  the  center  an  angle  of  133°. 

23.  What  is  the  angle  at  the  center  of  a circle  if  the  subtending 
chord  is  equal  to  § of  the  radius  ? 

24.  Find  the  area  of  a circular  sector  if  the  radius  of  the  circle  is 
12  in.,  and  the  angle  of  the  sector  is  30°. 

\\  25.  Find  the  tangent  of  the  angle  of  the  slope  of  an  A-roof  of  a 
building  which  is  24  ft.  6 in.  wide  at  the  eaves,  the  ridgepole  being 
10  ft.  9 in.  above  the  eaves. 


72 


PLAITE  TEIGOXOMETRY 


70.  The  Regular  Polygon.  We  have  already  considered  a few  cases 
involving  the  regular  polygon.  It  is  evident  from  geometry  that  if 
the  polygon  shown  below  has  n sides,  the  angle  of  the  right  triangle 
which  has  its  vertex  at  the  center  is  equal  to  J of  360°/n,  or  180°/n. 
The  triangle  may  evidently  be  solved  if  the  radius  of  the  circum- 
scribed circle  (r),  the  radius  of  the  inscribed  circle  (/i),  or  the  side  of 
the  polygon  (c)  is  given. 

In  the  exercises  we  shall  let 
n = number  of  sides, 
c = length  of  one  side, 
r = radius  of  circumscribed  circle, 
h = radius  of  inscribed  circle, 
p = the  perimeter, 

S = the  area. 

Then,  by  geometry, 

5 = ^ hp. 

Exercise  33.  The  Regular  Polygon 

Find  the  remaining  parts  of  a regular  polygon,  given : 

1.71=10,0=1.  3.  71  = 20,  ?•  = 20.  5.  77=11,  S' = 20. 

2.  77=18,  7’=  1.  4.  77  = 8,  77  =1.  6.  77  =7,  S=T. 

7.  The  side  of  a regular  inscribed  hexagon  is  1 in.;  find  the  side 
of  a regular  inscribed  dodecagon. 

\J  8.  Given  n and  c,  and  represent  by  b the  side  of  the  regular 
inscribed  polygon  having  2 n sides,  find  b in  terms  of  n and  c. 

9.  Compute  the  difference  between  the  areas  of  a regular  octagon 
and  a regular  nonagon  if  the  perimeter  of  each  is  16  in. 

10.  Compute  the  difference  between  the  perimeters  of  a regular 
pentagon  and  a regular  hexagon  if  the  area  of  each  is  12  sq.  in. 

11.  Find  the  perimeter  of  a regular  dodecagon  circumscribed  about 
a circle  the  circumference  of  which  is  1 in. 

12.  AlHiat  is  the  side  of  the  regular  inscribed  polygon  of  100  sides, 
the  radius  of  the  circle  being  unity  ? What  is  the  perimeter  ? 

is.  What  is  the  perimeter  of  the  regular  inscribed  polygon  of 
360  sides,  the  radius  of  the  circle  being  unity  ? 

■ 14.  The  area  of  a regular  polygon  of  twenty-five  sides  is  40  sq.  in ; 
find  the  area  of  the  ring  included  between  the  circumferences  of  the 
inscribed  and  circumscribed  circles. 


THE  EIGHT  TEIANGLE 


73 


Exercise  34.  Review  Problems 

1.  Prove  that  the  area  of  the  parallelogram  here  shown  is  equal 
to  ab  sin  A. 

2.  Two  sides  of  a parallelogram,  are  5 in.  and 
6 in.  respectively,  and  their  included  angle  is 
82°  45'.  What  is  the  area  ? 

3.  Two  sides  of  a parallelogram  are  9 ft. 
and  12  ft.  respectively,  and  their  included  angle  is  74.5°.' 
the  area  ? 

4.  Each  side  of  a rhombus  is  7.35  in.,  and  one  angle  is  42°  27'. 
What  is  the  area  ? 

5.  The  area  of  a rhombus  is  250  sq.  in.,  and  one  of  the  angles 
is  37°  25'.  What  is  the  length  of  each  side  ? 

6.  A pole  BD  stands  on  the  top  of  a mound  BC. 

- Erom  a point  A the  angles  of  elevation  of  the  top  and 

foot  of  the  pole  are  60°  and  30°  respectively.  Prove 
that  the  height  of  the  pole  is  twice  the  height  of  the 
mound. 

7.  A ladder  38  ft.  long  is  resting  against  a wall.  The  foot  of  the 
ladder  is  7 ft.  2 in.  from  the  wall.  What  is  the  height  of  the  top  of 
the  ladder  above  the  ground  ? 

8.  Erom  a boat  1325  ft.  from  the  base  of  a vertical  cliff  the  angle 
of  elevation  of  the  top  of  the  cliff  is  observed  to  be  14°  30'.  Eind 
the  height  of  the  cliff. 

9.  On  the  top  of  a building  50  ft.  high  there  is  a flagstaff  BD 
Erom  a point  A on  the  ground  the  angles  of  elevation  of  B and  D 
are  30°  and  45°  respectively.  Eind  the  length  of 
the  flagstaff  and  the  distance  AC  of  the  observer 
from  the  building,  as  shown  in  the  annexed  figure. 

50  50  I "u 

Since  — = tan  30°  and = tan  45°,  x can  evidently 

X X 

be  eliminated. 

10.  A man  whose  eye  is  5 ft.  8 in.  above  the  ground  stands  midway 
between  two  telegraph  poles  which  are  200  ft.  apart.  The  elevation 
of  the  top  of  each  pole  is  48°  50'.  What  is  the  height  of  each  ? 

11.  The  captain  of  a ship  observed  a lighthouse  directly  to  the 
east.  After  sailing  north  2 mi.  he  observed  it  to  lie  55°  30'  east  of 
south.  How  far  was  the  ship  from  the  lighthouse  at  the  time  of  each 
observation  ? 


74 


PLANE  TKIGONOMETRY 


12.  A leveling  instrument  is  placed  at  A on  the  slope  MN,  and  the 
line  M'N'  is  sighted  to  two  upright  rods.  By  measurement  MM'  is 
found  to  be  12.8  ft.,  NN'  to  be  3.4  ft.,  and  M'N'  to 

be  48.3  ft.  Bequired  the  angle  of  the  slope  of  MN 
and  the  distance  MN.  ^ 

13.  A wire  stay  is  fastened  to  a telegraph  pole  6.8  ft.  from  the 
ground  and  is  stretched  tightly  so  as  to  reach  the  ground  5.2  ft.  from 
the  foot  of  the  pole.  What  angle  does  the  wire  stay  make  with 
the  ground  ? 

14.  The  top  of  a conical  tent  is  8 ft.  7 in.  above  the  ground,  and 
the  diameter  of  the  base  is  9 ft.  8 in.  Find  the  inclination  of  the 
side  of  the  tent  to  the  horizontal.  Check  the  result  by  drawing  the 
figure  to  scale  and  measuring  the  angle  with  a protractor. 

16.  In  this  piece  of  iron  construction  work  AC  = 11  in.  and 
AB  makes  an  angle  of  30°  with  BC.  What  is  the  length  of  ^C? 

16.  In  Ex.  15  it  is  also  known  that  BE  and  CD 
are  each  9 in.  long  and  make  angles  of  60°  with  BC 
produced.  What  is  the  length  of  ED  ? 

17.  From  the  conditions  given  in  Ex.  16,  find  the 
length  of  CF. 

18.  The  base  of  a rectangle  is  14|  in.  and  the  diag- 
onal is  19^  in.  What  angle  does  the  diagonal  make  with  the  base  ? 
Check  the  result  by  drawing  the  figure  to  scale  and  measuring  the 
angle  with  a protractor. 

19.  In  constructing  the  spire  represented  in  the  figure  below  it  is 
planned  to  have  AA  = 42ft.  and  PM  =92  ft.  What  angle  of  slope 
must  the  builders  give  to  AP  ? 

20.  In  Ex.  19  find  the  length  of  AP  and  find  the 
angle  P. 

21.  In  the  figure  of  Ex.  19  the  brace  CD  is  put  in 
38  ft.  above  AB.  What  is  its  length  ? 

22.  The  spire  of  Ex.  19  rests  on  a tower.  A man 
standing  on  the  ground  at  a distance  of  400  ft.  from  the 
base  of  the  tower  observes  the  angle  of  elevation  of  P to  be  25°  38', 
the  instrument  being  5 ft.  above  the  ground.  What  is  the  height  of 
P above  the  ground  ? 

23.  When  the  angle  of  elevation  of  the  sun  is  38.4°,  what  is  the 
length  of  the  shadow  of  a tower  175  ft.  high  ? 


'U  ^ u 


THE  EIGHT  TEIAHGLE 


75 


24.  Two  men,  M and  N,  3200  ft.  apart,  observe  an  aeroplane  A 
at  the  same  instant,  and  at  a time  when  the  plane  MNA  is  vertical. 
The  angle  of  elevation  at  M is  41°  27'  and  the 
angle  at  N is  61°  42'.  Eequired  AB,  the  height  of 
the  aeroplane. 

Show  that  h cot  41°  27'  + h cot  61°  42'  is  known,  whence 
h can  be  found. 

25.  A kite  string  475  ft.  long  makes  an  angle  of  elevation  of 
49°  40'.  Assuming  the  string  to  be  straight,  what  is  the  altitude 
of  the  kite  ? 


26.  A steel  bridge  has  a truss  ADEF  in  which  it  is  given  that 
AD  = 20  ft.,  6 ft.  8 in.,  and  EA  =12  ft.,  as  ^ ^ 

shown  in  the  figure.  Eequired  the  angle  of  slope 
which  AF  makes  with  AD. 


A B 


D 


27.  Two  tangents  are  drawn  from  a point  P to  a 
circle  and  contain  an  angle  of  37.4°.  The  radius  of  the  circle  is  5 in. 
Eind  the  length  of  each  tangent  and  the  distance  of  P from  the  center. 


28.  Erom  the  top  of  a cliff  95  ft.  high,  the  angles  of  depression 
of  two  boats  at  sea  are  observed,  by  the  aid  of  an  instrument  5 ft. 
above  the  gromid,  to  be  45°  and  30°  respectively.  The  boats  are  in  a 
straight  line  with  a point  at  the  foot  of  the  cliff  directly  beneath  the 
observer.  What  is  the  distance  between  the  boats  ? 


29.  A carpenter’s  square  ^PCA/is  held  against  the  vertical  stick 
BD  resting  on  a sloping  roof  AD,  as  in  the  figure.  It  is  found 
that  AC  =24  in.  and  CP  = 11.5  in.  Eind  the 
angle  of  slope  of  the  roof  with  the  horizontal.  ^ 


30.  In  Ex.  29  find  the  length  of  AD. 


31.  A man  6 ft.  tall  stands  4 ft.  9 in.  from  a 

street  lamp.  If  the  length  of  his  shadow  is  19  it.,J 
how  high  is  the  light  above  the  street  ? ^ 

32.  The  shadow  of  a city  building  is  observed 
to  be  100  ft.  long,  and  at  the  same  time  the  shadow 
of  a lamp-post  9 ft.  high  is  observed  to  be  5.2  ft.  long.  Eind  the 
angle  of  elevation  of  the  sun  and  the  height  of  the 


building. 


33.  A man  5 ft.  10  in.  tall  walks  along  a straight  line  that  passes 
at  a distance  of  2 ft.  9 in.  from  a street  light.  If  the  light  is  9 ft. 
6 in.  above  the  ground,  find  the  length  of  the  man’s  shadow  when 
his  distance  from  the  point  on  his  path  that  is  nearest  to  the  lamp 
is  3 ft.  8 in. 


PLANE  TRIGONOMETKY 


76 


34.  A man  on  a bridge  35  ft.  above  a stream,  using  an  instrument 
5 ft.  high,  sees  a rowboat  at  an  angle  of  depression  of  27°  30'.  If 
the  boat  is  approaching  at  the  rate  of  2|  mi.  an  hour,  in  how  many 
seconds  will  it  reach  the  bridge  ? 


35.  A shaft  0,  of  diameter  4 in.,  makes  480  revolutions 
per  minute.  If  the  point  P starts  on  the  horizontal  line  OA . 
how  far  is  it  above  OA  after  of  a second? 


36.  Assrmring  the  earth  to  be  a sphere  with  radius  3957  mi.,  find 
the  radius  of  the  circle  of  latitude  which  passes  through  a place  in 


latitude  47°  27'  10"  N. 


37.  When  a hoisting  crane  AB,  28  ft.  long,  makes  an  angle  of 
23°  with  the  horizontal  A C,  what  is  the  length  of  A C ? Suppose 

that  the  angle  CAB  is  doubled,  what  is  then  a— 

the  length  of  A C ? | 

38.  In  Ex.  37  find  the  length  of  BC  in  I 

each  of  the  two  cases.  'o 

39.  Wishing  to  measure  the  distance  AB,  a man  swings  a 100-foot 
tape  line  about  B,  describing  an  arc  on  the  ground,  and  then  does  the 
same  about  A.  The  arcs  intersect  at  C,  and  the 
angle  ACB  is  found  to  be  32°  10'.  What  is  the 
length  of  AP  ? 

40.  From  the  top  of  a mountain  15,250  ft.  high, 
overlooking  the  sea  to  the  south,  over  how  many  minutes  of  latitude 
can  a person  see  if  he  looks  southward  ? Use  the  assumption  stated 
in  Ex.  36. 


41.  The  length  of  each  blade  of  a pair  of  shears,  from  the  screw 
to  the  point,  is  5^-  in.  When  the  points  of  the  open  shears  are  3|-  in. 
apart,  what  angle  do  the  blades  make  with  each  other  ? 

42.  Ill  Ex.  41  how  far  apart  are  the  points  when  the  blades  make 
an  angle  of  28°  45'  with  each  other  ? 

43.  The  wheel  here  represented  has  eight  spokes, 
each  being  19  in.  long.  How  far  is  it  from  A to  P ? 
from  P to  P ? 

^ 44.  The  angle  of  elevation  of  a balloon  from  a 
station  directly  south  of  it  is  60°.  From  a second  station  lying 
5280  ft.  directly  west  of  the  first  one  the  angle  of  elevation  is  45°. 
The  instrument  being  5 ft.  above  the  level  of  the  ground,  what  is 
the  height  of  the  balloon  ? 


CHAPTER  V 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE 

71.  Need  for  Oblique  Angles.  We  have  thus  far  considered  only- 
right  triangles,  or  triangles  which  can  readily  be  cut  into  right  tri- 
angles for  purposes  of  solution.  There  are,  however,  oblique  triangles 
which  cannot  conveniently  be  solved  by  merely  separating  them  into 
right  triangles.  We  are  therefore  led  to  consider  the  functions  of 
oblique  angles,  and  to  enlarge  our  idea  of  angles  so  as  to  include 
angles  greater  than  180°,  angles  greater  than  360°,  and  even  negative 
angles  and  the  angle  0°. 


72.  Positive  and  Negative  Angles.  We  have  learned  in  algebra  that 
we  may  distinguish  between  two  lines  which  extend  in  opposite  direc- 
tions by  calling  one  positive  and  the  other  negative. 

For  example,  in  the  annexed  figure  we  consider  OX  T 

as  positive  and  therefore  OX'  as  negative.  We  also  con-  + 

sider  OY  as  positive  and  hence  OY'  as  negative.  In  gen-  _ 

eral,  horizontal  lines  extending  to  the  right  of  a point  Xr q 'X 

which  we  select  as  zero  are  considered  positive,  and  those 

to  the  left  negative.  Vertical  lines  extending  upward  from 

zero  are  considered  positive,  and  those  extending  down-  x 

ward  are  considered  negative. 


With  respect  to  angles,  an  angle  is  considered  positive  if  the  rotat- 
ing line  which  describes  it  moves  counterclockwise,  that  is,  in  the 
direction  opposite  to  that  taken  by  the  hands  of  a 
clock.  An  angle  is  considered  negative  if  the  rotat- 
ing line  moves  clockwise,  that  is,  in  the  same 
direction  as  that  taken  by  the  hands  of  a clock. 

Arcs  which  subtend  positive  angles  are  considered 
positive,  and  arcs  which  subtend  negative  angles 
are  considered  negative.  Thus  /.A OB  and  arc  AB  are  considered 
positive;  Z.AOB'  and  arc  AB'  are  considered  negative. 


For  example,  we  may  think  of  a pendulum  as  swinging  through  a positive 
angle  when  it  swings  to  the  right,  and  through  a negative  angle  when  it  swings 
to  the  left.  We  may  also  think  of  an  angle  of  elevation  as  positive  and  an  angle 
of  depression  as  negative,  if  it  appears  to  be  advantageous  to  do  so  in  the  solu- 
tion of  a problem. 


77 


78 


PLANE  TEIGONOMETKY 


73.  Coordinates  of  a Point.  In  trigonometry,  as  in  work  with 
graphs  in  algebra,  we  locate  a point  in  a plane  by  means  of  its 
distances  from  two  perpendicular  lines. 

These  lines  are  lettered  XX'  and  YY',  and  their  point  of  intersection  0. 
The  lines  are  called  the  axes  and  the  point  of  intersection  the  origin. 

In  some  branches  of  mathematics  it  is  more  convenient  to  use  oblique  axes, 
but  in  trigonometry  rectangular  axes  are 
used  as  here  shown. 

The  distance  of  any  point  P from 
the  axis  XX' , or  the  a;-axis,  is  called 
the  ordinate  of  the  point.  Its  distance 
from  the  axis  YY',  or  the  y-axis,  is 
called  the  abscissa  of  the  point. 

In  the  figure,  y is  the  ordinate  of  P,  and 
X is  the  abscissa  of  P.  The  point  P is  rep- 
resented by  the  symbol  (x,  y).  In  the  figure 
the  side  of  each  small  square  may  be  taken 
to  represent  one  unit,  in  which  case  P = (4,  3),  because  its  abscissa  is  4 and 
its  ordinate  3.  Following  a helpful  European  custom,  the  points  are  indicated 
by  small  circles,  so  as  to  show  more  clearly  when  a line  is  drawn  through  them. 

The  abscissa  and  ordinate  of  a point  are  together  called  the  coordi- 
nates of  the  po  int. 

74.  Signs  of  the  Coordinates.  Erom  § 73  we  see  that  odiscissas  to  the 
right  of  the  y-axis  are  positive  ; abscissas  to  the  left  of  the  y-axis  are 
negative  ; ordinates  above  the  x-axis  are  positive  ; ordinates  below  the 
x-axis  are  negative. 

A point  on  the  line  YY'  has  zero  for  its  abscissa,  and  hence  the  abscis.sa  may 
be  considered  as  either  positive  or  negative  and  may  be  indicated  by  ± 0.  Simi- 
larly, a point  on  the  line  XX'  has  ± 0 for  its  ordinate. 

75.  The  Four  Quadrants.  The  axes  divide  the  plane  into  four  parts 
known  as  quadrants. 

Because  angles  are  generally  considered  as  generated  by  the  rotating  line 
moving  counterclockwise,  the  four  quadrants  are  named  in  a counterclockwise 
order.  Quadrant  XOY  is  spoken  of  as  the  first  quadrant,  YOX'  as  the  second 
quadrant,  X'OY'  as  the  third  quadrant,  and  Y'OX  as  the  fourth  quadrant. 

76.  Signs  of  the  Coordinates  in  the  Several  Quadrants.  Erom  § 74 
we  have  the  following  rule  of  signs : 

In  quadrant  I the  abscissa  is  qoositive,  the  ordinate  positive  ; 

In  quadrant  II  the  abscissa  is  negative,  the  ordinate  positive  ; 

In  quadrant  III  the  abscissa  is  negative,  the  ordinate  negative  ; 

In  quadrant  IV  the  abscissa  is  positive,  the  ordinate  negative. 


P(2 


V) 


X' 


■Y'- 


FUNCTIONS  OF  ANY  ANGLE 


79 


77.  Plotting  a Point.  Locating  a point,  having  given  its  coordi- 
nates, is  c,?il\edL  plotting  the  point. 


L, 

1 

(- 

) 

I 

X 

X 

1 

d 

n 

(- 

3r 

8) 

I 

rt 

F- 

0 

■M 

P 

0 

0 

(1 

1) 

Xi 

Y 

jr 

0 

rr 

2 

■/ 

0 

■S 

< 

to; 

a ' 

Por  example,  in  the  first  of  these  figures  the  point  (—2,  4)  is  shown  in 
quadrant  II,  the  point  (—  3,  — 2)  in  quadrant  III,  and  the  point  (1,  — 1)  in 
quadrant  IV. 

In  the  second  figure  the  point  (—  2,  0)  is  shown  on  OX',  between  quad- 
rants II  and  III,  and  the  point  (1,  0)  on  OX,  between  quadrants  I and  IV. 

In  the  third  figure  the  point  (0, 1)  is  shown  on  OY,  between  quadrants  I and  II, 
and  the  point  (0,  — 3)  on  OY',  between  quadrants  III  and  IV. 

In  every  case  the  origin  0 may  be  designated  as  the  point  (0,0). 


78.  Distance  from  the  Origin.  The  coordinates  of  P being  x and 
y,  we  may  form  a right  triangle  the  hypotenuse  of  which  is  the 
distance  of  P from  0. 

Kepresenting  OP  by  r,  we  have 


?■  = V; 


Since  this  may  be  written  r = ± ~Vx^  + y^,  we  see  that 
r may  be  considered  as  either  positive  or  negative.  It  is 
the  custom,  however,  to  consider  the  rotating  line  which 
forms  the  angle  as  positive.  If  r is  produced  through  0, 
the  production  is  considered  as  negative. 

1.  What  is  the  distance  of  the  point  (3,  4)  from  the  origin  ? 

r = V3F+42  = = 5. 


2.  What  is  the  distance  of  the  point  (—  3,  — 2)  from  the  origin  ? 

r = V(-  3)2  -1-  (-  2)2  = V9  -I-  4 = Vl3  = 3.61. 


3.  What  is  the  distance  of  the  point  (5,  — 5)  from  the  origin  ? 

r = V52  + (-  5)2  = VEo  = 7.07. 


4.  What  is  the  distance  of  the  point  (—  2,  0)  from  the  origin  ? 
r = V(-  2)2  + 02  = \/i  = 2, 
as  is  evident  from  the  conditions  of  the  problem. 


80 


PLANE  TEIGONOMETRY 


Exercise  35.  Distances  from  the  Origin 

Using  squared  paper  ^ or  measuring  with  a ruler  ^ plot  the  follow- 
ing points : 


1. 

(2,  3). 

8. 

(- 

-3, 

2). 

15. 

(3, 

-4). 

22. 

(0,  0). 

2. 

(3,  5). 

9. 

(- 

-3, 

4). 

16. 

(4, 

-3). 

23. 

(0,  24). 

3. 

(4,  4). 

10. 

(- 

-5, 

!)• 

17. 

(5, 

-!)• 

24. 

(0,  - 34). 

4. 

(2i,  3). 

11. 

(- 

-4, 

6). 

18. 

(0, 

-)• 

25. 

(44,  0). 

5. 

(3i,  44). 

12. 

(- 

-2). 

19. 

(3, 

0). 

26. 

(54,  0). 

6. 

(4i,  44). 

13. 

(- 

-3, 

-5). 

20. 

(0, 

-4). 

27. 

(-  24,  0). 

7. 

(5i,  34). 

14. 

(- 

- 5, 

-3). 

21. 

(- 

2,  0). 

28. 

o' 

CO 

Find  the  distance  of  each  of  the  following  points  from  the  origin: 

29.  (6,  8).  32.  (1^,  2).  35.  (2,  Vs).  38.  (0,  7). 

30.  (9,  12).  33.  (I,  1).  36.  (-  3,  4).  39.  (5,  0). 

31.  (5,12).  34.  37.  (0,0).  ^ 40.  (-12,-9). 

41.  Find  the  distance  from  (3,  2)  to  (—  2,  3). 

42.  Find  the  distance  from  (—  3,  — 2)  to  (2,  — 3). 

43.  Find  the  distance  from  (4,  1)  to  (—  4,  —1). 

V;44.  Find  the  distance  from  (0,  3)  to  (—  3,  0). 

45.  A point  moves  to  the  right  7 in.,  up  4 in.,  to  the  right  10  in., 
and  up  18f  in.  How  far  is  it  then  from  the  starting  point  ? 

46.  A point  moves  to  the  right  9 in.,  up  5 in.,  to  the  left  4 in.,  and 
up  3 in.  How  far  is  it  then  from  the  starting  point  ? 

47.  Find  the  distance  from  (—  4^3)  to  (^,  — 4 Vs). 

48.  A triangle  is  formed  by  joining  the  points  (1,  0),  (—  4,  4 

and  (—  — 4'^)-  Find  the  perimeter  of  the  triangle.  Draw  the 

figure  to  scale. 

49.  Find  the  area  of  the  triangle  in  Ex.  48. 

U 50.  A hexagon  is  formed  by  joining  in  order  the  points  (1,  0), 
(i,iV3),  (-^,^V3),  (-1,0),  (-^,-^V3),  (i, -^V3),  and 
(1,  0).  Is  the  figure  a regular  hexagon  ? Prove  it. 

51.  A polygon  is  formed  by  joining  in  order  the  points  (1,  0), 

(iV2,  iV2),  (0,  1),  (-^V2,iV2),  (-1,  0),  (-^V2,  -i^), 

(0,  — 1),  (■4-  V2,  — 4-  V2),  and  (1,  0).  Draw  the  figure,  state  the  kind 
of  polygon,  and  find  its  area. 


FUNCTIONS  OF  AKY  A^sGLE 


81 


79.  Angles  of  any  Magnitude.  In  the  following  figures,  if  the  rotat- 
ing line  OP  revolves  about  0 from  the  position  OX,  in  a counterclock- 
wise direction,  until  it  again  coincides  with  OX,  it  will  generate  all 
angles  in  every  quadrant  from  0°  to  360°. 

The  line  OX  is  called  the  initial  side  of  the  angle,  and  the  line  OP  the  ter- 
minal side  of  the  angle. 


An  angle  is  said  to  be  an  angle  of  that  quadrant  in  which  its 
terminal  side  lies. 


Angles  between  0°  and  90°  are  angles  of  quadrant  I. 

Angles  between  90°  and  180°  are  angles  of  quadrant  II. 

Angles  between  180°  and  270°  are  angles  of  quadrant  III. 

Angles  between  270°  and  360°  are  angles  of  quadrant  IV. 


The  rotating  line  may  also  pass  through  360°,  forming  angles  from 
360°  to  720°.  It  may  then  make  another  revolution,  forming  angles 
greater  than  720°,  and  so  on  in- 
definitely. 


For  example,  in  using  a screwdriver 
we  turn  through  angles  of  360°,  720°, 

1080°,  and  so  on,  depending  upon  the 
number  of  revolutions.  In  the  same  way, 
the  minute  hand  of  a clock  turns  through  8640°  in  a day,  and  the  drive  wheel 
of  an  engine  may  turn  through  thousands  of  degrees  in  an  hour. 

We  might,  if  necessary,  speak  of  an  angle  of  400°  as  an  angle  of  quadrant  I, 
because  its  terminal  side  is  in  that  quadrant,  but  we  have  no  occasion  to  do  so 
in  practical  cases. 


As  stated  in  § 72,  if  the  line  OP  is  rotated  clockwise,  it  generates 
negative  angles. 

In  this  way  we  may  form  angles  of  — 40°  or  — 140°,  as  here  shown,  and  the 
rotation  may  continue  until  we  have  angles  of  — 360°,  — 720°,  — 1080°,  — 1440°, 
and  so  on  indefinitely. 

We  shall  have  but  little  need  for  the 
negative  angle  in  the  practical  work 
of  trigonometry,  but  we  shall  make  ex- 
tensive use  of  angles  between  0°  and 
180°,  and  some  use  of  those  between 
180°  and  360°. 


/ H 

i \ 

I ^ 

P<(ni 

"ivy 

82 


PLANE  TPIGONOMETKY 


80.  Functions  of  Any  Angle.  Since  we  have  now  seen  that  we  may 
have  angles  of  any  magnitude,  it  is  necessary  to  consider  their  func- 
tions. Although  we  must  define  these  functions  anew,  it  will  be 
seen  that  the  definitions  hold  for  the  acute  angles  which  we  have 
already  considered. 


In  whatever  quadrant  the  angle  is,  we  designate  it  by  A.  We  take 
a point  P,  or  (x,  y),  on  the  rotating  line,  and  let  OP  = r.  Then  the 
angle  XOP,  read  counterclockwise,  is  the  angle  A.  We  then  define 
the  functions  as  follows  : 


. , y ordinate 

sin  A = - = -r— , 

r distance 

, X abscissa 

cosA=  - = , 

r distance 

, , y ordinate 

tanA=  -=  ^ , 

X abscissa 


CSC  A = 


sec  A - 


cot  A 


1 distance 

sin  A y ordinate  ’ 

1 distance 

cos  A X abscissa 

1 _ abscissa 

tan  A y ordinate 


It  will  be  seen  that  these  definitions  are  practically  the  same  as  those  already 
learned  for  angles  in  quadrant  I.  Their  application  to  the  other  quadrants 
is  apparent.  The  general  definitions  might  have  been  given  at  first,  but  this 
plan  offers  difficulties  for  a beginner  which  make  it  undesirable. 

By  counting  the  squares  on  squared  paper  and  thus  getting  the  lengths  of 
certain  lines,  the  approximate  values  of  the  functions  of  any  given  angle  may  be 
found,  but  the  exercise  has  no  practical  significance.  The  values  of  the  functions 
are  determined  by  series,  these  being  explained  in  works  on  the  calculus. 


FUNCTIONS  OF  ANY  ANGLE 


83 


81.  Angles  determined  by  Functions.  Given  any  function  of  an 
angle,  it  is  possible  to  construct  the  angle  or  angles  which  satisfy 
the  value  of  the  function. 


1.  Given  sinA  = f,  construct  the  angle  A. 

If  we  take  a line  parallel  to  X'X  and  3 units  above  it,  and  then  rotate  a 
line  OP,  5 units  long,  about  0 until  P rests  upon  this  parallel,  we  shall  have 


In  other  words,  we  have  constructed  two  angles,  each  of  which  has  3-  for 
its  sine. 

Furthermore,  we  could  construct  an  infinite  number  of  such  angles,  for  we 
see  that  360°  + A terminates  in  OP  and  has  the  same  sine  that  A has,  and  that 
tue  same  may  be  said  of  360°  + A',  720°  + A,  720°  + A',  1080°  + A,  and  so  on. 

In  general,  therefore,  the  angle  n x 360°  + A has  the  same  functions  as  A,  ?i 
being  any  integer.  Hence  if  we  know  the  value  of  any  particular  function  of 
an  angle,  the  angle  cannot  be  uniquely  determined  ; that  is,  there  is  more  than 
one  angle  which  satisfies  the  condition.  In  general,  as  we  see,  an  infinite  number 
of  angles  will  satisfy  the  given  condition,  although  this  gives  no  trouble  because 
only  two  of  these  angles  can  be  less  than  360°. 


2.  Given  tanA=  construct  the  angle  A. 

If  we  take  an  abscissa  4 and  an  ordinate  3,  as  in  quadrant  I of  the  figure, 
we  locate  the  point  (3,  4).  Then  angle  XOP  has  for  its  tangent  But  it  is 
evident  that  we  may  also  locate  the  point  (—  3,  — 4)  in  quadrant  III,  and  thus 
find  an  angle  between  180°  and  270°  whose  tangent  is 

82.  Functions  found  from  Other  Functions.  Given  any  function  of  an 
angle,  it  is  possible  not  only  to  construct  the  angle  but  also  to  find 
the  other  functions. 


For  in  Ex.  1 above,  after  constructing  angles  A and  A',  we  see  that 


sin  A = 

5 

cos  A = - or ■, 

5 5 

tan  A = - or , 

4 - 4 


cscA  = 


^ 5 5 

sec  A = - or , 

4 - 4 

. 4 - 4 

cot  A = - or 

3 3 


That  is,  if  sin  A = |^,  then  cos  A = ± tan  A = ± f,  esc  A = sec  A = ± 
and  cot  A = ± |. 


84 


PLANE  TRIGONOMETRY 


Exercise  36.  Construction  of  Angles  and  Functions 

Using  the  protractor,  construct  the  following  angles: 

1.  30°.  4.  150°.  7.  270°.  10.  405°.  13.  -45°. 

2.  60°.  6.  180°.  8.  300°.  11.  450°.  14.  - 90°. 

3.  80°.  6.  200°.  9.  360°.  12.  720°.  15.  - 180°. 


State  the  quadrants  in  which  the  terminal  sides  of  the  following 
angles  lie : 

16.  45°.  19.  150°.  22.  390°.  25.  660°.  28.  930°. 

17.  75°.  20.  210°.  23.  495°.  26.  765°.  29.  990°. 


18.  120°.  21. 

315°.  24.  570°.  27.  820°. 

30.  1080°. 

Construct  two  angles  A,  given  the  following . 

\/  31.  sin  A = 

36.  sin  A = — |. 

41. 

sin  A = — 1. 

32.  cos  A= 

v/  37.  cosA  = — !■. 

42. 

cos  A = — 1. 

33.  tan  A 

\/38.  tanA  = — 

43. 

tanA  = — 1. 

34.  cot  A = ^. 

39.  cot  A = — 

1/44. 

cot  A = — 1. 

35.  sec  A=  2. 

40.  sec  A = — 1. 

46. 

sec  A = — 2. 

Given  the  following  functions  of  angle  A,  construct  the  other 
functions : 


46. 

sin  A = 

2 

T- 

61. 

sin  A = - 

-t- 

56. 

sin  A = — 

47. 

cos  A - 

3 

52. 

cos  A-- 

-1. 

57. 

cos  A = — 

48. 

tanA  = 

4 

X- 

63. 

tan  A = - 

_ 3 

S- 

58. 

tan  A——h. 

49. 

cot  A = 

3 

64. 

sec  A = - 

- 2. 

59. 

cot  A = — ^. 

60. 

CSC  A = 

3. 

65. 

CSC  A = - 

-1. 

60. 

sec  A = — 24. 

61.  If  tan4  = V2,  show  that  cotT  is  half  as  large.  What  are  the 
values  of  sinA,  cosA,  secA,  and  cscA  ? 

62.  The  product  2 sin  45°  cos  45°  is  equal  to  the 
sine  of  what  angle? 

63.  The  product  2 sin  30°  cos  30°  is  equal  to  the 
sine  of  what  angle  ? 

64.  To  the  diagonal  A C of  a square  A BCD  a perpendicular  AM 
is  drawn.  Find  the  values  of  the  six  functions  of  angle  BAM. 

65.  In  the  figure  of  Ex.  64,  suppose  AM  rotates  further,  until  it 
is  in  line  with  BA.  What  are  then  the  six  functions  of  angle  BAM? 


FUNCTIONS  OF  ANY  ANGLE 


85 


83.  Line  Values  of  the  Functions.  As  in  the  case  of  acute  angles 
(§  22)  we  may  represent  the  trigonometric  functions  of  any  angle 
by  means  of  lines  in  a circle  of  radius  unity. 

Thus  in  each  of  the  following  figures 


sin  X = MP, 
cos  X = OM, 


tan  X — AT, 
cot  X - BS, 


seex  = OT, 
CSC  X = OS. 


By  examining  the  figures  we  see  that 
In  quadrant  I all  the  functions  are  positive ; 

In  quadrant  II  the  sine  and  cosecant  only  are  positive ; 

In  quadrant  III  the  tangent  and  cotangent  only  are  positive ; 
In  quadrant  IV  the  cosine  and  secant  only  are  positive. 


It  will  be  seen  as  we  proceed  that  the  laws  and  relations  which 
have  been  found  for  the  functions  of  acute  angles  hold  for  the  func- 
tions of  angles  greater  than  90°.  For  example,  it  is  apparent  from 
the  above  figures  that,  in  every  quadrant, 

MP"  + = 1, 

and  hence  that  sinM  + cosM  = 1, 
as  shown  in  § 14.  It  is  also  evident  that 

AT  _ ^ 

1 ~ om’ 

sin.4 

and  hence  that  tan^  = -• 

cosJ. 


Other  similar  relations  are  easily  proved  by  reference  to  the  figures. 


86 


PLAXJirTRIGONOMETRY 


84.  Variations  in  the  Functions.  A study  of  the  line  values  of 
the  functions  shows  how  they  change  as  the  angle  increases  from 
0°  to  360°. 

1.  The  Sine.  In  the  first  quadrant  the  sine  MP 
is  positive,  and  increases  from  0 to  1 ; in  the 
second  it  remains  positive,  and  decreases  from 
1 to  0 ; In  the  third  it  is  negative,  and  increases 
in  absolute  value  from  0 to  1 ; in  the  fourth  it 
is  negative,  and  decreases  in  absolute  value  from 
1 to  0.  The  absolute  value  of  the  sine  varies, 
therefore,  from  0 to  1,  and  its  total  range  of  values  is  from  + 1 to  — 1. 

In  the  third  quadrant  the  sine  decreases  from  0 to  — 1,  but  the  absolute  value 
(the  value  without  reference  to  its  sign)  increases  from  0 to  1,  and  similarly 
for  other  cases  on  this  page  in  which  the  absolute  value  is  mentioned. 

2.  The  Cosine.  In  the  first  quadrant  the  cosine  OM  is  positive, 
and  decreases  from  1 to  0 ; in  the  second  it  becomes  negative,  and 
increases  in  absolute  value  from  0 to  1 ; in  the  third  it  is  negative, 
and  decreases  in  absolute  value  from  1 to  0 ; in  the  fourth  it  is 
positive,  and  increases  from  0 to  1.  The  absolute  value  of  the 
cosine  varies,  therefore,  from  0 to  1. 

3.  The  Tangent.  In  the  first  quadrant  the  tangent  A T is  positive, 
and  increases  from  0 to  oo  ; in  the  second  it  becomes  negative,  and 
decreases  in  absolute  value  from  oo  to  0 ; in  the  third  it  is  positive, 
and  increases  from  0 to  oo  ; in  the  fourth  it  is  negative,  and  decreases 
in  absolute  value  from  oo  to  0. 

4.  The  Cotangent.  In  the  first  quadrant  the  cotangent  BS  is  posi- 
tive, and  decreases  from  oo  to  0;  in  the  second  it  is  negative,  and 
increases  in  absolute  value  from  0 to  oo  ; in  the  third  and  fourth  quad- 
rants it  has  the  same  sign,  and  undergoes  the  same  changes  as  in  the 
first  and  second  quadrants  respectively.  The  tangent  and  cotangent 
may  therefore  have  any  values  whatever,  positive  or  negative. 

5.  The  Secant.  In  the  first  quadrant  the  secant  Or  is  positive,  and 
increases  from  1 to  oo  ; in  the  second  it  is  negative,  and  decreases  in 
absolute  value  from  oo  to  1 ; in  the  third  it  is  negative,  and  increases 
in  absolute  value  from  1 to  oo  ; in  the  fourth  it  is  positive,  and  decreases 
from  00  to  1. 

6.  The  Cosecant.  In  the  first  quadrant  the  cosecant  OS  is  positive, 
and  decreases  from  oo  to  1 ; in  the  second  it  is  positive,  and  increases 
from  1 to  00  ; in  the  third  it  is  negative,  and  decreases  in  absolute 
value  from  oo  to  1;  in  the  fourth  it  is  negative,  and  increases  in 
absolute  value  from  1 to  oo. 


FUNCTIONS  OF  ANY  ANGLE 


87 


It  is  evident,  therefore,  that  the  sine  can  never  be  greater  than  1 
nor  less  than  — 1,  and  that  it  has  these  limiting  values  at  90°  and 
270°  respectively.  We  may  also  say  that  its  absolute  value  can  never 
be  greater  than  1,  and  that  it  has  its  limiting  value  0 at  0°  and  180°, 
and  its  limiting  absolute  value  1 at  90°  and  270°. 

If  we  have  an  equation  in  which  the  value  of  the  sine  is  found  to  he  greater 
than  1 or  less  than  — 1,  we  know  either  that  the  equation  is  wrong  or  that  an 
error  has  been  made  in  the  solution. 

Of  course  the  values  of  the  functions  of  360°  are  the  same  as  those  of  0°, 
since  the  moving  radius  has  returned  to  its  original  position  and  the  initial  and 
terminal  sides  of  the  angle  coincide. 

In  the  same  way,  the  absolute  value  of  the  cosine  cannot  be  greater 
than  1,  and  it  has  its  limiting  value  0 at  90°  and  270°,  and  its  limit- 
ing absolute  value  1 at  0°  and  180°.  Similarly  we  can  find  the 
limiting  values  of  all  the  other  functions. 

For  convenience  we  speak  of  oo  as  a limiting  value,  although  the  function 
increases  without  limit,  the  meaning  of  the  expression  in  this  case  being  clear. 

Summarizing  these  results,  we  have  the  following  table : 


Function 

0° 

90° 

180° 

270° 

360° 

Sine 

y 0 

+ 1 

±0 

-1 

y 0 

Cosine 

y 1 

±0 

-1 

y 0 

+ 1 

Tangent 

y 0 

± CO 

y 0 

± oo 

y 0 

Cotangent 

y CO 

±0 

y CO 

±0 

y 00 

Secant 

y 1 

± CO 

-1 

y 00 

y 1 

Cosecant 

y CO 

y 1 

± 00 

- 1 

y CO 

\ 

Sines  and  cosines  vary  in  value  from  +1  to  — 1 ; tangents  and  cotangents, 
from  -h  00  to  — 00  ; secants  and  cosecants,  from  -f  oo  to  -f  1,  and  from  — 1 to  — oo  . 

In  the  table  given  above  the  double  sign  ± or  y is  placed  before  0 and  oo  . 
From  the  preceding  investigation  it  appears  that  the  functions  always  change 
sign  in  passing  through  0 or  through  oo  ; and  the  sign  ± or  y prefixed  to  0 or  oo 
simply  shows  the  direction  from  which  the  value  is  reached.  For  example,  at  0° 
the  sine  is  passing  from  — (in  quadrant  IV)  to  y (in  quadrant  I).  At  90°  the 
tangent  is  passing  from  y (in  quadrant  I)  to  — (in  quadrant  II). 

85.  Functions  of  Angles  Greater  than  360°.  The  functions  of  360°  -t-  x 
are  the  same  in  sign  and  in  absolute  value  as  tdiose  of  x.  If  n is  a 
pc^itive  integer. 

The  funotions  af  (n  x 360°  -(-  x)  are  the  sarm  as  those  of  x. 

For  example,  the  functions  of  2200°,  or  6 x 360°  y 40°,  are  the  same  in  sign 
and  in  absolute  value  as  the  functions  of  40°. 


88 


PLANE  TEIGONOMETKY 


Exercise  37.  Variations  in  the  Functions 


Represent  the  following  functions  hy  lines  in  a unit  circle  : 


1.  sin  135°. 

2.  cos  120°. 

3.  tan  150°. 

4.  cot  135°. 

5.  sec  120°. 

6.  CSC  150°. 


7.  sin  210°. 

8.  cos  225°. 

9.  tan  240°. 

10.  cot  210°. 

11.  sec  225°. 

12.  CSC  240°. 


13.  sin  300°. 

14.  cos  315°. 

15.  tan  330°. 

16.  cot  300°. 

17.  sec  315°. 

18.  CSC  330°. 


19.  sin  270°. 

20.  cos  180°. 

21.  tan  180°. 

22.  cot  270°. 

23.  sec  180°. 

24.  CSC  270°. 


25.  Prepare  a table  showing  the  signs  of  all  the  functions  in 
each  of  the  four  quadrants. 

26.  Prepare  a table  showing  which  functions  always  have  the 
minus  sign  in  each  of  the  four  quadrants. 


Represent  the  following  functions  by  lines  in  a unit  circle : 

27.  sin  390°.  30.  cos  390°.  33.  sin  460°.  36.  tan  475°. 

28.  tan  405°.  31.  cot  405°.  34.  sin  570°.  37.  sec  705°. 

29.  sec  420°.  32.  esc  420°.  35.  sin  720°.  38.  esc  810°. 


Show  by  lines  in  a unit  circle  that 

39.  sin  150°  = sin  30°. 

40.  cos  150°  = — cos  30°. 

41.  sin  210°  = - sin  30°. 

42.  COS  210°  = — cos  30°. 

43.  sin  330°  = — sin  30°. 

44.  cos  330°=  cos  30°. 


45.  tan  120°  = — tan  60°. 

46.  cot  120°  = — cot  60°. 

47.  tan  240°  = tan  60°. 

48.  cot  240°  = cot  60°. 

49.  tan  300°  = — tan  60°. 
y^50.  cot  300°  = — cot  60°. 


51.  Write  the  signs  of  the  functions  of  the  following  angles: 
340°,  239°,  145°,  400°,  700°,  1200°,  3800°. 


52.  How  many  values  less  than  360°  can  the  angle  x have  if 
sin  a;  = + f , and  in  what  quadrants  do  the  angles  lie  ? Draw  a figure. 

53.  How  many  values  less  than  720°  can  the  angle  x have  if 
cos  a:  = + f , and  in  what  quadrants  do  the  angles  lie  ? Draw  a figure. 

54.  If  we  take  into  account  only  angles  less  than  180°,  how  many 
values  can  x have  if  sin  a;  = f ? if  cos  x = \l  if  cos  x = — %?  if 
tan  a;  = § ? if  cot  a:  = — 7 ? 

55.  Within  what  limits  between  0°  and  360°  must  the  angle  x lie 
if  cos  X =—  I ? if  cot  X = 4 ? if  sec x = 80  ? if  esc x = — 3 ? 


FUNCTIONS  OF  ANY  ANGLE 


89 


66.  Why  may  cot  360°  be  considered  as  either  + oo  or  — oo  ? 

57.  Find  the  valnes  of  sin  460°,  tan  540°,  cos  630°,  cot  720°,  sin810° 

CSC  900°,  cos  1800°,  sin  3600°. 

58.  What  functions  of  an  angle  of  a triangle  may  be  negative  ? 

In  what  cases  are  they  negative  ? 

\X 59.  In  what  quadrant  does  an  angle  lie  if  sine  and  cosine  are  both 
negative  ? if  cosine  and  tangent  are  both  negative  ? 

60.  Between  0°  and  3600°  how  many  angles  are  there  whose  sines 
have  the  absolute  value  f ? Of  these  sines  how  many  are  positive  ? 

Compute  the  values  of  the  following  expressions : '\ 

61.  a sin  0°  + ^ cos  90°  — c tan  180°. 

62.  a cos  90°  — h tan  180°  + c cot  90°. 

63.  a sin  90°  — h cos  360°  -\-(a  — b)  cos  180°. 

V 64.  (a^  — 6^)  cos  360°  — 4 sin  270°  + sin  360°. 

65.  cos  180°+  (a"  + 1“^)  sin  180°  + (a"  + tan  135°. 

66.  (a^  + 2a6  + 5^)sin90°  +(a^  — 2ab-\-  6^)cos  180°—  4aS  tan225°. 

67.  (a  — b c — c^sin  210° —(a  — b + c — d')GOS,  180°  + a tan  360°. 

State  the  sign  of  each  of  the  six  functions  of  the  following  angles : 

68.  75°.  70.  155°.  72.  275°.  74.  355°. 

69.  125°.  71.  185°.  73.  325°.  75.  - 66°. 

Find  the  four  smallest  angles  that  satisfy  the  following  conditions : 

76.  sind  = ^.  78.  sin.4  = -|V3.  80.  tan^  = -jV3. 

77.  cos^=|-V3.  79.  cosd.=  -^.  v^81.  tan^=V3. 

Find  two  angles  less  than  360°  that  satisfy  the  following  conditions : 

82.  sinJ.  = -4- 84.  sind  = — -^V^.  86.  tan.4  = — 1. 

83.  cos^  = — I-.  85.  cosd  = — ^ V2.  87.  cotd  = — 1. 

iff  A,  B,  and  C are  the  angles  of  any  triangle  AB  C*,  prove  that : 

88.  cos -l-d.  = sin-^(5  + C).  90.  cos = sin + C). 

89.  sin-|-C=  cos-^(d +5).  91.  sin-|-d.=  cos-^(iJ  + C). 

As  angle  A increases  from  0°  to  360°,  trace  the  changes  in  sign 
and  magnitude  of  the  following : 

92.  sin d,  cos d-.  94.  sin^  — cos^.  96.  tand  + cotA. 


93.  sinA  + cosA.  95.  sinA-^cosA. 


90 


PLANE  TPIGONOMETPY 


86.  Reduction  of  Functions  to  the  First  Quadrant.  In  tlie  annexed 
figure  BB'  is  perpendicular  to  the  horizontal  diameter  AA  \ and  the 
diameters  PR  and  QS  are  so  drawn  as  to  B 

make  Z.AOP—  Z.SOA.  It  therefore  fol- 
lows from  geometry  that  A MOP,  MOS, 

NOQ,  and  NOR  are  congruent. 

Considering,  therefore,  only  the  absolute 
values  of  the  functions,  we  have 

sin  A OP  = sin  AOQ  = sin  A OR,  = sin  A OS, 
cos  AOP  = cos  A OQ  = cos  = cos^O.?, 

and  so  on  for  the  other  functions. 

Hence,  For  every  acute  angle  there  is  an  angle  in  each  of  the  h igher 
quadrants  whose  functions,  in  absolute  value,  are  equal  to  those  of 
this  acute  angle. 

If  we  let  AAOP  = x and  Z.POB  = y,  noticing  that  Z.AOP  — 
ZQOA'=  ZA'OR  = ZSOA  = X,  and  Z.POB  ^ FBOQ  ^ Z.ROB’ = 
Z.B'OS  = y,  and  prefixing  the  proper  signs  to  the  functions  (§  83), 
we  have : 


Angle  in  Quadrant  II 


^ sin  (180°  — x)=  sin  x 
^ cos  (180°  — x)  = — cos  X 
V tan  (180°  — x)  = — tan  x 
cot  (180°  — a;)  = — cot  x 


sin  (90°  r y}=  cos  y 
cos  (90°  -f  ?/)  = — sin  y 
tan  (90°  -f  y)  = — cot  y 
cot  (90°  -p  y)  = — tan  y 


Angle  in 
sin  (180°  -p  x)  — — sin  x 
cos  (180°  -\-F)  ——  cos  X 
tan  (180°  -p  a;)  = tan  x 
cot  (180°  -p  a:)  = cot  x 

Angle  in 


Quadrant  III 

sin  (270°  — y)  ——  cos  y 
cos  (270°  — y)  = — sin  y 
tan  (27 0°  — y)  = cot  y 
cot  (270°  — y)  = tany 

Quadrant  IV 


sin  (360°  — x)  =—  sin  x sin  (270°  + y)  = — cos  y 

cos  (360°  — x)  = cos  X cos  (270°  -P  y)  = sin  y 

tan  (360°  — x)  = — tan  x tan  (270°  -f  y)  - — cot  y 

cot  (360°  — x)  = — cot  X cot  (270°  -P  y)  = — tan  y 

For  example,  sin  127°  = sin  (180°  — 53°)  = sin  53°  = cos  37°, 

sin  210°  = sin  (180°  -p  30°)  = — sin  30°  = — cos  60°, 
sin  350°  = sin  (360°  — 10°)  = — sin  10°  = — cos  80°. 


and 


FUNCTIONS  OF  ANY  ANGLE 


91 


It  appears  from  the  results  set  forth  on  page  90  that  the  functions 
of  any  angle,  however  great,  can  he  reduced  to  the  functions  of  an 
angle  in  the  first  quadrant. 

Por  example,  suppose  that  we  have  a polygon  with  a reentrant  angle  of 
247°  30',  and  we  wish  to  find  the  tangent  of  this  angle.  We  may  proceed  by 
finding  tan  (180°  + x)  or  by  finding  tan  (270°  — x).  We  then  have 

tan  247°  30'  = tan  (180°  + 67°  30')  = tan  67°  30', 
and  tan  247°  30'  = tan  (270°  - 22°  30')  = cot  22°  30'. 

That  these  two  results  are  equal  is  apparent,  for 

tan  67°  30'  = cot  (90°  - 67°  30')  = cot  22°  30'. 

It  also  appears  that,  for  angles  less  than  180°,  a given  value  of  a 
sine  or  cosecant  determines  two  swpjplementary  angles,  one  acute,  the 
other  obtuse  ; a given  value  of  any  other  function  determines  only  one 
angle,  this  angle  being  acute  if  the  value  is  positive  and  obtuse  if  the 
value  is  negative. 

For  example,  if  we  know  that  sin  x = ^,  we  cannot  tell  whether  x = 30°  or 
160°,  since  the  sine  of  each  of  these  angles  is  But  if  we  know  that  tanx  = 1, 
we  know  that  x = 45°. 

Similarly,  if  we  know  that  cot  x = — 1,  we  know  that  x = 135°,  there  being 
no  other  angle  less  than  180°  whose  cotangent  is  — 1. 

Since  sec  x is  the  reciprocal  of  cos  x and  esc  x is  the  reciprocal  of  sin  x,  and 
since  by  the  aid  of  logarithms  we  can  divide  by  cos  x or  sin  x as  easily  as  we 
can  multiply  by  sec  x or  esc  x,  we  shall  hereafter  pay  but  little  attention  to  the 
secant  and  cosecant.  Since  the  invention  of  logarithms  these  functions  have 
been  of  little  practical  importance  in  the  work  of  ordinary  mensuration. 


Exercise  38.  Reduction  to  the  First  Quadrant 


Express  the  following  as  functions  of  angles  less  than  90°  : 


1. 

sin  170°. 

11. 

sin  275°. 

21. 

sin  148° 

10' 

2. 

cos  160°. 

12. 

sin  345°. 

22. 

cos  192° 

20' 

r 

3. 

tan  148°. 

13. 

tan  282°. 

23. 

tan  265° 

30 

t 

4. 

cot  156°. 

14. 

tan  325°. 

24. 

cot  287° 

40' 

sin  180°. 

^15. 

cos  290°. 

{^26. 

sin  187° 

10' 

3". 

6. 

tan  180°. 

16. 

cos  350°. 

26. 

cos  274° 

5' ; 

14". 

7. 

sin  200°. 

17. 

cot  295°.^ 

27. 

tan  322° 

8' 

15". 

8. 

cos  225°. 

18. 

cot  347°. 

28. 

cot  375° 

10' 

3". 

9. 

tan  258°. 

19. 

sin  360°. 

29. 

sin  147.75°. 

^ 10. 

cot  262°. 

^20. 

cos  360°. 

1/30. 

cos  232.25°. 

92 


PLANE  TKIGONOMETPY 


87.  Functions  of  Angles  Differing  by  90°.  It  was  shown  in  the  case 
of  acute  angles  that  the  function  of  any  angle  is  equal  to  the  co-func- 
tion of  its  complement  (§  8). 

That  is,  tan  28°  = cot  (90°  — 28°)  = cot  62°, 
sinx  =:  cos  (90°  — x),  and  so  on. 

It  will  now  be  shown  for  all  angles 
that  if  two  angles  differ  hy  90°,  the  func- 
tions of  either  are  equal  in  absolute  value 
to  the  co-functions  of  the  other. 

In  the  annexed  figure  the  diameters  PR 
and  Q.S  are  perpendicular  to  each  other, 
and  from  P,  Q,  R,  and  S perpendiculars  are  drawn  to  AA\  Then 
from  the  congruent  triangles  OMP,  QHO,  OKR,  and  SNO  we  see  that 


and 


OM=  QH=  OK  = SN, 
MP  = OH  = KR  - - ON. 


Hence,  considering  the  proper  signs  (§  83), 

sin^OQ  = cosAOP,  cos^  OQ  = — sinAOP, 
sinfi  OP  = cosAOC^,  cos  A OP  =— sin  A OQ, 

sin  A 08’  = cos  A OP,  cos  A 08  = — sin  A OP. 

In  all  these  equations,  if  x denotes  the  angle  on  the  right-hand 
side,  the  angle  on  the  left-hand  side  is  90°  -f  x. 

Therefore,  if  x is  an  angle  in  any  one  of  the  four  quadrants, 
sin  (90°  -|-  x)  = cos  x,  cos  (90°  -|-  x)  = — sin  x ; 
and  hence  tan  (90°  -|-  x)  = — cot  x,  cot  (90°  -}-  x)  = — tan  x. 

It  is  therefore  seen  that  the  algebraic  sign  of  the  function  of  the  resulting 
angle  is  the  same  as  that  found  in  the  similar  case  in  § 86. 


88.  Functions  of  a Negative  Angle.  If  the  angle  x is  generated 
by  the  radius  moving  clockwise  from  the  initial  position  OA  to  the 
terminal  position  OS,  it  will  be  negative  (§  72),  and  its  terminal 
side  will  be  identical  with  that  for  the  b 

angle  360°  — x.  Therefore  the  functions 
of  the  angle  — x are  the  same  as  those 
of  the  angle  360°  — x ; or 

sin  (—  x)  — sin  x, 
cos  (—  x)  = cos  X, 
tan  (—  x)  = — tan  x, 
cot  (—  x)  = — cot  X. 


FUNCTIONS  OF  ANY  ANGLE 


93 


Exercise  39.  Reduction  of  Functions 


Express  the  following  as  functions  of  angles  less  than  45°  : 


1.  sin  100°. 

2.  sin  120°. 

3.  sin  110°. 

4.  sin  130°. 


5.  cos  95°. 

6.  cos  97°. 

7.  cos  111°. 

8.  cos  127°. 


9.  tan  91°. 

10.  tan  99°. 

11.  tan  119°. 

12.  tan  129°. 


Express  the  following  as  functions  of  positive  angles  : 


17.  sin  (—3°). 

18.  sin  (—9°). 

19.  sin(—  86°). 

20.  cos  (—75°). 


21.  cos(—  87°). 

22.  cos  (—95°). 

23.  tan  (—100°). 

24.  tan  (—150°). 


13.  cot  94°  1’. 

14.  cot  97°  2'. 

15.  cot  98°  3'. 
y 16.  cot  99°  9'. 

9 

25.  tan(—  200°). 

26.  cot (—1.5°). 

27.  cot(-7.8°). 
1/28.  cot  (— 9.1°). 


Find  the  following  hy  aid  of  the  tables : 


29. 

sin  178°  30'. 

37. 

log  sin  127.5°. 

30. 

cos  236°  45', 

38. 

log  cos  226.4°. 

31. 

tan  322°  18' 

39. 

log  tan  327.8°. 

32. 

cot  423°  15'. 

40. 

log  cot  343.3°. 

33. 

sin  (-7°  29 

' 30"). 

41. 

log  sin  236°  13 

'5". 

34. 

cos  (-  29°  42'  19"). 

42. 

log  cos  327°  5' 

11". 

36. 

tan  (-172° 

16'  14"). 

43. 

log  tan  (— 125° 

27'). 

36. 

cot  (—  262° 

17'  15"). 

44. 

log  cot  (—  236‘ 

' 15'). 

45. 

Show  that  ■ 

the  angles  42°,  138°, 

- 318 

°,  402°,  and  - 

222° 

have  the  same  sine. 

46.  Find  four  angles  between  0°  and  720°  which  satisfy  the  equa- 
tion sin  x=—  ^ V2. 

47.  Draw  a circle  with  unit  radius,  and  represent  by  lines  the 
sine,  cosine,  tangent,  and  cotangent  of  — 325°. 

48.  Show  by  drawing  a figure  that  sin  195°  = cos  (— 105°),  and 
that  cos  300°  = sin  (—  210°). 

^49.  Show  by  drawing  a figure  that  cos  320°  = — cos  (—140°),  and 
that  sin  320°  = — sin  40°. 

50.  Show  by  drawing  a figure  that  sin  765°  = J V2,  and  that 
tan  1395°  = -!. 

51.  In  the  triangle  ABC  show  that  cosd  = — cos  (A -|- C),  and 
that  cos  B — — cos  (A  -f  C). 


94 


PLANE  TRIGONOMETRY 


89.  Relations  of  the  Functions.  Certain  relations  between  tbe  func- 
tions have  already  been  proved  to  exist  in  the  case  of  acute  angles 
(§§  13,  14),  and  since  the  relations  of  the  functions  of  any  angle  to 
the  functions  of  an  acute  angle  have  also  been  considered  (§§  80,  85, 
86,  88),  it  is  evident  that  the  laws  are  true  for  any  angle.  These 
laws  are  so  important  that  they  will  now  be  summarized,  and  others 
of  a similar  kind  will  be  added. 

These  laws  should  be  memorized.  They  will  be  needed  frequently  in  the 
subsequent  work.  The  proof  of  each  should  be  given,  as  required  in  § 14. 
The  ± sign  is  placed  before  the  square  root  sign,  since  we  have  now  learned 
the  nooning  of  negative  functions. 

To  find  the  sine  we  have : 

1 

sm  X — 

CSC  X 

To  find  the  cosine  we  have : 

1 

cos  X = 

sec  a: 


sin  X = ± Vl—  cos^x 


cos  X = ± V 1 — sin^x 


To  find  the  tangent  we  have : 


tan  X = — ; — 
cot  X 

, sinx 

tan  X = ± —i=== 
V 1 — sin^x 


tan  X = ± Vsec^x  — 1 


To  find  the  cotangent  we  have : 


cot  X - 


tan  X 


cot  X — ± 


cos  X 


VT 


cos“x 


cot  X = ± Vcsc^x  — 1 


To  find  the  secant  we  have : 

1 

sec  X = 

cos  X 


To  find  the  cosecant  we  have : 

1 

CSCX  = 

sin  X 


, sinx 

tanx  = 

cos  X 

, Vl  — cos^x 

tanx  = ± I- 

COSX  iCi_  I 

tan  X = sin  x sec  x 

f ..  ^ 


cos  X 

cot  X = 

Sin  X 


, Vl  — siVx 

cot  X = + ^ 

sm  X 


cot  X = cos  X CSC  X 


sec  x = ± Vl  -f-  taVx 


CSC  X — ± Vl-|-  cot^x 


FUNCTIONS  OF  ANY  ANGLE 


95 


Exercise  40.  Relations,  of  the  Functions 

1.  Prove  each,  of  the  formulas  given  in  § 89. 

Prove  the  following  relations  : 

2.  sin  a; 


± 


3.  cos  X = ± 


tana: 


Vl  + tan^a: 

cot  a: 


Vl+  cot^a; 

6.  Find  sin  x in  terms  of  cot  x. 

7.  Find  cos  x in  terms  of  tan  x. 

Prove  the  following  relations : 

10.  tan  X cos  x = sin  x. 

11.  cos^a;  = cot^a;  — cot^a:  cos^a:. 

12.  tw?x  — sin^a:  + sin^x  tan^x. 

13.  cos^x  + 2sin^x  =1  + sin^x. 


4.  tan  X = ± 

5.  cot  X = ± 


V csc^x  — 1 

1 


Vsec^x  —1 

8.  Find  sec  x in  terms  of  sin  x. 

9.  Find  esc  x in  terms  of  cos  x. 


14.  cot^x  = cos^x  + cos^x  coUx. 

15.  cot^x  sec^x  = 1 + cot"x. 

16.  csc^x  — cot^x  = 1. 

17.  sec^x  + csc^x  = sec^x  csc^x. 


y 18.  Show  that  the  sum  of  the  tangent  and  cotangent  of  an  angle 
is  equal  to  the  product  of  the  secant  and  cosecant  of  the  angle. 

Recalling  the  values  given  on  page  8,  find  the  value  of  x when : 

19.  2cosx  = secx,  25.  tanx  = 2sinx. 

20.  4sinx  = cscx.  26.  sec  x = V2  tan  x. 

21.  sin^x  = 3 eos^x.  ■ '27.  sin^x  — cos  x = 

22.  2 sin^x  + cos^x  = f.  28.  tan"x  — secx  = 1. 

23.  3 tan^x  — sec'^x  = 1.  29.  tan^x  + csc'^x  = 3. 

24.  tan  x + cot  x = 2.  M 30.  sin  x + VS  cos  x = 2. 

'^31.  Given  (sin  x + cos  x)^  — 1 = (sin  x — cos  x)‘^  + 1,  find  x. 

32.  Given  2 sin  x = cos  x,  find  sin  x and  cos  x. 

33.  Given  4 sin  x = tan  x,  find  sin  x and  tan  x. 

34.  Given  5 sin  x = tan  x,  find  cos  x and  sec  x. 

35.  Given  4cotx  tanx,  find  the  other  functions. 

36.  Given  sin  x = 4 cos  x,  find  sin  x and  cos  x. 

37.  If  sin  X : cos  x = 9 : 40,  find  sinx  and  cos  x. 

sinx 


38.  From  the  formula  tanx  = ± 
under  which  tan  x = sin  x. 


Vl 


find  the  condition 


sm'x 


96 


PLANE  TEIGONOMETPY 


Solve  the  following  equations  ; that  is,  find  the  value  of  x when : 


44.  2 cos  X + sec  a:  = 3. 

45.  cos^a:  — sin^a:  = sin  x. 

46.  2 sin  a:  + cot  a:  = 1 + 2 cos  x. 

47.  sin^a;  + tan^x  = 3 cos^x. 

48.  tan  X + 2 cot  x = f esc  x. 


39.  cos  X = seex. 

40.  cos  X = tan  x. 

41.  cos  X = sin  X. 

42.  tan  X = cot  X. 

43.  sec  X = esex. 

Prove  the  following  relations  : 

49.  sin^ +COS  (l-|-tanyl)cos4.  51.  cos x ; cot x = Vl  — cos'^ x. 


50. 


cotx 


= Vl  + cot'^ 


52.  taVx 


cos  X ■ cos^x 

Find  the  values  of  the  other  functions  of  A when: 


-1. 


53.  sin  ^ = f. 

54.  cos  ^ = f . 

55.  tanyl  = 1.6. 

56.  cot  = 0.75. 

57.  sec^  = 1.5. 


58.  sin^  = 

59.  sinyl  = 0.8. 

60.  cos^  = fy. 

61.  cos  .4  =0.28. 

62.  tan  = f. 


63.  cot^  = l. 

64.  cot^l=0.5. 

65.  sec  A=2. 

66.  CSC  A = V2. 

67.  sin  .4  - m. 


68.  Given  sin  ^ = 2 m : (1  + find  the  value  of  tand. 

69.  Given  cos  A = 2 mn : (rrd‘  + V),  find  the  value  of  sec.4. 

7 0.  Given  sin  0°  = 0,  find  the  other  functions  of  0°. 

71.  Given  sin  90°  = 1,  find  the  other  functions  of  90°. 

72.  Given  tan  90°  = oo,  find  the  other  functions  of  90°. 

73.  Given  cot  22°  30'  = V2  + 1,  find  the  other  functions  of  22°  30'. 

74.  Write  taV^  + cot^d  so  as  to  contain  only  cosd. 

In  the  triangle  ABC,  prove  the  following  relations  : 

75.  sin  d = sin(£  + C).  83.  sin  d = — eos(|d+.^54-.lC). 

76.  cos  d = — cos  (A  + C).  84.  cosd  = — cos(2  d + .B  + C). 

77.  tand  =— tan(5+ C).  85.  cosd  = sin(fd  + 1-5  + 

78.  cotd  = — cot(5  + C).  86.  sin  (-Id  4-5)  = cos (-15  — -I C). 

79.  sin  d =— sin  (2d  4-54- C).  87.  sin(l-C— Id)  =— cos  (-1-54-5). 

80.  sin  5 = — sin (d  4-254- C).  88.  cos5  = — cos(d  4- 2 5 4- C). 

81.  cos  5 = — cos (d  4-54-25).  89.  tand  = tan(2  d 4- 5 4- 5). 

82.  cot  5 = cot  (d  4-2  5 4- 5).  90.  cot  d = tan  (|-54-f  5 4- 1-d). 


In  the  quadrilateral  ABCD,  prove  the  following  relations  : 

91.  — sin d = sin (5  4- 5 4- 5).  93.  — tand  = tan (5  4- 5 4- 5). 

92.  cos d=  cos (5  4- 5 4- 5).  94.  — cotd  = cot(5  4- 5 -f -D)- 


CHAPTER  VI 


FUNCTIONS  OF  THE  SUM  OR  THE  DIFFERENCE  OF  TWO  ANGLES 


90.  Formula  for  sin(jf  + y).  In  this  figure  there  are  shown  two 

acute  angles,  x and  y,  with  Z.AOC  acute  and  equal  to  x-\-y,  two 

perpendiculars  are  let  fall  from  C,  and  two  from  D,  as  shown.  Then 

by  geometry  the  triangles  CGD  and  EOD  are  similar  and  hence 

Z.GCD  - Z.EOD  = X.  Considering  the  radius  as  unity,  OD  = cos  y 

and  CD  - sin  y.  Hence  we  have 

sin  (x  A-y)=  CF  = DE  + CG. 

^ . DE  , 

But  sin  X = j whence  DE  = sin  x-  OD 
OD 

= sin  X cos  y ; 

CG  , 

and  cos  x = > whence  CG=  cos  x ■ CD 

= cos  X sin  y. 

Hence  sin  (x-\-y)  = sin  jt  cos  y + cos  x sin  y. 

This  is  one  of  the  most  important  formulas  and  should  he  memorized. 

For  example,  sin  (30°  + 60°)  = sin  30°  cos  60°  + cos  30°  sin  60° 

Vs  Vs 

which  we  have  already  found  to  he  sin  90°. 


1 1 


_ 1 3_ 

2 ’ 


91.  Formula  for  cos  (x-\-  y).  Using  the  above  figure  we  see  that 
cos  (x  y)=  OF  = OE  — DG. 

OE 

But  cos  X — , whence  OE  = cos  x ■ OD  = cos  x cos  y ; 

DG 

and  sin  x = , whence  DG  = sin  x- CD  = sin  x sin  y. 

Hence  cos  (jr  + y)  = cos  x cos  y — sin  x sin  y. 

This  important  formula  should  he  memorized. 

For  example,  cos  (45°  + 45°)  = cos  45°  cos  45°  — sin  45°  sin  45° 

_ 1 ^ 

V2  V2  V2  V2  2 2 

which  we  have  already  found  to  he  cos  90°. 

97 


98 


PLANE  TKIGONOMETRY 


92.  The  Proofs  continued.  In  the  proofs  given  on  page  97,  x,  y, 
and  X + y were  assumed  to  be  acute  angles.  If,  however,  x and  y 
are  acute  but  x + y is  obtuse,  as  shown  in 
this  figure,  the  proofs  remain,  word  for 
word,  the  same  as  before,  the  only  differ- 
ence being  that  the  sign  of  OF  will  be  nega- 
tive, as  Ziff  is  now  greater  than  OE.  This,  ^ 
however,  does  not  affect  the  proof.  The 
above  formulas,  therefore,  hold  true  for  aU  acute  angles  x and  y. 

Furthermore,  if  these  formulas  hold  true  for  any  two  acute  angles 
X and  y,  they  hold  true  when  one  of  the  angles  is  increased  by  90°. 
Thus,  if  for  x we  write  x'  = 90°  -f-  x,  then,  by  § 87, 


sin  (x'  + y)  = sin  (90°  -j-  x + y)  = cos (x  + y) 

= cos  X cos  y — sin  x sin  y. 

But  by  § 87,  cos  x = sin  (90°  + x)  = sin  x\ 
and  sin  x = — cos  (90°  + x)  = — cos  x'. 

Hence,  by  substituting  these  values, 

sin  (x'  -t-  y)  = sin  x'  cos  y -f  cos  x'  sin  y. 

That  is,  § 90  holds  true  if  either  angle  is  repeatedly  increased  by  90°.  It  is 
therefore  true  for  all  angles. 


Similarly,  by  § 87, 

cos (x'  y)  = cos  (90°  + X -(-  y)  = — sin(x  + y) 

= — sin  X cos  y — cos  x sin  y 
= cos  x'  cos  y — sin  x'  sin  y, 

by  substituting  cos  x'  for  — sin  x and  sin  x'  for  cos  x as  above. 

That  is,  § 91  also  holds  true  if  either  angle  is  repeatedly  increased  by  90°. 
It  is  therefore  true  for  all  angles. 


Exercise  41.  Sines  and  Cosines 

Given  sin 30°  = cos 60°  = ^,  cos 30°  = sin 60°  = and  sin 43° 

— cos  45°  = j v^,  find  the  values  of  the  following : 


1.  sin  15°. 

2.  cos  15°. 

3.  sin  75°. 

4.  cos  75°. 


6.  sin  90°. 

6.  cos  90°. 

7.  sin  105°. 
V '8.  cos  105°. 


9.  sin  120°. 

10.  cos  120°. 

11.  sin  135°. 

12.  COS  135°. 


13.  sin  150'. 

14.  COS  150°. 

15.  sin  105°. 

16.  cos  166°. 


SUM  OR  UIRFERENCE  OF  TWO  ANGLES 


99 


93.  Formula  for  tan  (jr  + y).  Since  tan  A = 
tan  (x  y)  = 


sin  A 


therefore 


cos  A 

sin  (x  + y)  _ sin  x cos  y + cos  x sin  y 


cos  (x  + y)  cos  X cos  y — sin  x sin  y 

whatever  the  size  of  the  angles  x and  y 92). 

Dividing  each  term  of  the  numerator  and  denominator  of  the 
last  of  these  fractions  by  cos  x cos  y,  we  have 


tan  {x  y)  - 


sin  X , sin  y 

1 

cos  X cos  y 


1- 


sina;  siny 
cos  X cos  y 


But  since 
we  have 


sin  a;  , ^ sin?/ 

= tan  X,  and = tan  y, 

Gosx  cosy 


, . tan  X 4-  tan  y 

''  1 — tan  X tan  y 

This  important  formula  should  be  memorized. 


94.  Formula  for  cot  ('jr+y').  Since  cotA=  — -j  therefore 
^ ^ sm  A 

, , ^ cos  (x  4-  ?/)  cos  X cos  y — sin  x sin  y 

cot  (x  + 'll)  = -7—7 : — - > 

sin(a:  + y)  sin  x cos  y + cos  x sin  y 


whatever  the  size  of  the  angles  x and  y (§  92). 

Dividing  each  term  of  the  numerator  and  denominator  of  the 

last  of  these  fractions  by  sin  x sin  y,  and  then  remembering  that 

cos  X ^ ^ cos  y , , 

— = cot  a:  and  ^ = cot  y,  we  have 
sin  X sin  y 


cot(jf+y)  = 


cot  Jf  cot  y — 1 
cot  y + cot  a: 


This  important  formula  should  be  memorized. 


Exercise  42.  Tangents  and  Cotangents 


Given  tan  30°  = cot  60°  = | cot  30°  = tan  60°  = tan  45° 
= cot  45°  = i,  find  the  values  of  the  following : 


1.  tan  15°. 
y 2.  cot  15°. 

3.  tan  75°. 

4.  cot  75°. 


5.  tan  90°. 

6.  cot  90°. 

7.  tan  105°. 

8.  cot  105°. 


9.  tan  120°. 

10.  cot  120°. 

11.  tan  135°. 

12.  cot  135°. 


13.  tan  150°. 

14.  cot  150°. 
^15.  tanl65°i 

16.  cot  165° 


100 


PLANE  TKIGONOMETEY 


95.  Formula  for  sin  (x  — y).  In  this  figure  there  are  shown  two  acute 
angles,  A OB  = x and  COB  = y,  with  /LAOC  equal  tox  — y,  two  per- 
pendiculars are  let  fall  from  C,  and  two  from  D. 

The  perpendiculars  from  D are  BE  and  BG,  BG 
being  drawn  to  FC  produced. 

Then,  considering  the  radius  as  unity,  we  have 
sin  (x  — y)  = CF=  BE  - CG. 

But  DE  = sin  x • OD  = sin  x cos  y, 

and  GC  = cos  x • CD  = cos  x sin  y. 

Hence,  by  substituting  these  values  of  DE  and  GC, 
sin  (x—y)  = sin  xcosy  — cos  x sin  y. 

This  is  one  of  the  most  important  formulas  and  should  be  memorized. 

96.  Formula  for  cos  (x—y).  Using  the  above  figure  we  see  that 

cos  (x  — y)=  OF  — OE  -f-  DG. 

But  OE  — cos  X • OD  = cos  x cos  y,  ^ 

and  DG  = sin  x • CD  = sin  x sin  y. 

Hence  it  follows  that 

cos  (x  — y)z=  cos  jr  cos  -p  sin  x sin  y. 

This  important  formula  should  be  memorized.  The  proof  in  §§  95  and  96 
refers  only  to  acute  angles,  but  the  formulas  are  entirely  general  if  due  regard 
is  paid  to  the  algebraic  signs.  The  general  proof  may  follow  the  method  of 
§ 92,  or  it  may  be  based  upon  it;  the  latter  plan  is  followed  in  § 97. 

97.  The  Proofs  continued.  Since  cc  = (x  — y)  -p  y,  we  see  that 

sin  X = sin  {(x  — y)  + y}  = sin  (x  — y)  cos  y -p  cos  (x  — y)  sin  y, 

cos  X = cos  {(x  — y)  + y}  = cos  (x  — y)  cos  y — sin  (x  — y)  sin  y. 

Multiplying  the  first  equation  by  cos  y,  and  the  second  by  sin  y, 
sin  X cos  y = sin  (x  — y)  cos^y  -p  cos  (x  — y)  sin  y cos  y, 
cos  X sin  y = — sin  (x  — y)  sin^y  + cos  (x  — y)  sin  y cos  y. 

Hence  sin  x cos  y — cos  x sin  y = sin  (x  — y)  (sin^y  -p  cos'^y). 

But  by  § 14  sin^y  -p  cos^y  = 1. 

Therefore  sin  (x  — y)  = sin  x cos  y — cos  x sin  y. 

Similarly,  cos  (x  — y)  — cos  x cos  y + sin  x sin  y. 

Therefore  the  formulas  of  §§95  and  96  are  tmiversally  true. 


SUM  on  DIFFERENCE  OF  TWO  ANGLES  101 


sin  A 


98.  Formula  for  tan  (x  — y).  Since  tan /I  = 

' - ^ cos  A 

i/  tai.(»=-y)=Sl&^ 


we  have 


cos  {x  — y) 

_ sin  X cos  y — cos  x sin  y 
cos  X cos  y + sin  x sin  y 

Dividing  numerator  and  denominator  by  cos  x cos  y,  as  in  § 93,  we 
obtain  sin  x sin  y 

cos  X cos  y 


tan  (x  — y~)  = 


1 + 


sin  X sm  y 
cos  X cos  y 


m,  a.  • a.  / X tan  j:  — tan  1/ 

That  IS,  tan (x—y)  = . 

’ •'  l+tanxtanz/ 

This  important  formula  should  be  memorized. 


99.  Formula  for  cot  (x— jr). 

we  niav  show  that 

cot  (x  — y)  = 


Following  the  plan  suggested  in  § 98, 

cos  (x-  — ij) 
sin  (x  — y) 

cos  X cos  y -f-  sin  x sin  y 
sin  X cos  y — cos  x sin  y 


cos  X 
sin  X 
cos  y 
sin  7/ 


cosy 

sin  y 
cos  X 
sin  X 


That  is. 


cot  {x  — y)  = 


cot  xcotyA- 1 
cot  y —cot  X 


/ This  important  formula  should  be  memorized. 

100.  Summary  of  the  Addition  Formulas.  The  formulas  of  §§  90-99 
may  be  combined  as  follows  : 


sin  {x  ±_y)  — sin  a;  cos  y ±_  cos  a;  sin  y, 
cos  (a;  ± y)  = cos  x cos  y zf  sin  x sin  y, 


tan(a:  ± y)  = 


tan  X ± tan  y 
1 ^ tan  X tan  y 


cot  (x  ±y)^ 


cot  X cot  y ^ 1 
cot  y ± cot  X 


When  the  signs  ± and  occur  in  the  same  formula  we  should  be  careful  to 
take  the  — of  T with  the  + of  ± . That  is,  the  upper  signs  are  to  be  taken 
together,  and  the  lower  signs  are  to  be  taken  together. 


102 


PLANE  TRIGONOMETRY 


Exercise  43.  The  Addition  Formulas 

G-iven  sinx  = j,  cosx  = j^,  siny  = cosy  = find  the  value  of: 

1.  sm(£c  + y).  3.  cos(cc  + y).  5.  tan(x  + ?/). 

2.  sin  {x  — y).  4.  cos  {x  — y).  6.  tan  (a;  — y). 

By  letting  x = 90°  in  the  formulas,  find  the  following  : 

7.  sin(90°-?/).  8.  cos(90°-?/).  9.  tan(90°-y). 


Similarly,  hy  substituting  in  the  formulas,  find  the  following , 


10.  sin(90'’ + ?/). 

11.  sin  (180°  — y) 

12.  sin(180°  + ?/) 

13.  sin  (270°  — y) 

14.  sin(270°  + y) 

15.  sin(360°  — ?/) 

16.  sin(360°  + y) 


24.  sin  (—  y). 

25.  sin  (45°  — y). 

26.  cos  (45°  — y). 

27.  tan(45°  — y), 

28.  cot(30°  + y). 

29.  cot  (60°  — y). 

30.  cot  (90°  — y). 


17.  cos  {x  — 90°). 

18.  cos  (x  — 180°). 

19.  cos  (x  — 270°). 

20.  tan  (x  — 90°). 

21.  tan  (a:  — 180°). 

22.  cot  (x  — 90°). 

23.  cot  (x  — 180°). 

31.  If  tan  X = 0.5  and  tan  y = 0.25,  find  tan  (x  + y)  and  tan  (x  — y) 

32.  If  tan  a:  = 1 and  tan  y = ^ Vs,  find  tan  (x  + y)  and  tan(a;  — y). 

33.  If  tan  x — ^ and  tan  y = -^,  find  tan  (x  + y)  and  tan  (x  — y), 
and  find  the  number  of  degrees  in  a:  + y. 

34.  If  tana:  = 2 and  tant/  = what  is  the  nature  of  the  angle 
X y?  Consider  the  same  question  when  tan  x — d and  tan  y = \, 
and  when  tan  x = a and  tan  y = 1/a. 

35.  Prove  that  the  sum  of  tan  (x  — 45°)  and  cot  (x  + 45°)  is  zero. 

36.  Prove  that  the  sum  of  cot  (x  — 45°)  and  tan  (x  + 45°)  is  zero. 

37.  If  sin  X = 0.2  Vs  and  sin  y — 0.1  VTo,  prove  that  x-\-y  = 45° 
May  a:  + y have  other  values  ? If  so,  state  two  of  these  values. 

38.  Prove  that  if  an  angle  x is  decreased  by  45°  the  cotangent  of 
the  resulting  angle  is  equal  to  — a;  -f- 1 _ 

39.  Prove  that  if  an  angle  x is  increased  by  45°  the  cotangent  of 

the  resulting  angle  is  equal  to  — r’ 

cot  a:  + 1 

^ and  tan  y = ^ > prove  that  tan  (a;  + y)  = 1. 


40.  If  tana:  = 


1 a 


l+2a 


41.  If  a righr  angle  is  divided  into  any  three  angles  x,  y, 
1 — tan  y tan  g 


prove 


that  tan  x = 


tan  y + tan  z 


SUM  OE  DIFFERENCE  OF  TWO  ANGLES  103 


101.  Functions  of  Twice  an  Angle.  By  substituting  in  the  formulas 
for  the  functions  of  x + y we  obtain  the  following  important  for- 
mulas for  the  functions  of  twice  an  angle : 

sin  2 jr  = 2 sin  x cos  x, 
cos  2 jr  = cos^  X — sin*^  x, 


tan  2 x = 


2 tan  jc 
1 — tan^  X ’ 


cot  2 jr  = 


cot^  Jf  — 1 
2 cot  Jf 


Letting  2x  = y we  have  the  following  useful  formulas  : 
sin  y = 2 sin  ^ y cos  y, 


cos  y = cos'^ — sin^ ^ y, 


tany  = 


2 tan  ^ y 
1 — tan^  i y 


coty  = 


cot^  \y  — 1 

2 cot  ^ 1/ 


Exercise  44.  Functions  of  Twice  an  Angle 

As  suggested  above,  deduce  the  formulas  for  the  following : 

1.  sin  2 a;.  2.  cos  2 a:.  3.  tan  2 a;.-  4.  cot  2 a:. 

Find  sin2x,  given  the  following  values  of  sin  x and  cosx: 

5.  sin  a;  = ^ V2,  cos  a;  = V2.  6.  sin  a:  = cos  a:  = -I" 

Find  cos  2 x,  given  the  following  values  of  sin  x and  cos  x : 

7.  sin  x = \ Vs,  cos  x = ^.  8.  sin  a;  = f , cos  a;  = 

Find  tan  2 x,  given  the  following  values  of  tan  x : 

9.  tan  X = 0.3673.  10.  tan  x = 0.2701. 

Find  cot  2 x,  given  the  following  values  of  cot  x and  tan  x : 

11.  cota;  = 0.3673.  12.  tan  a;  = 0.2701. 

Find  sin  2 x,  given  the  following  values  of  sin  x : 

13.  sina:  = .j^.  14.  sina:  = ^. 

16.  As  suggested  in  § 101,  find  sin  3 a:  in  terms  of  sin  a:. 

16.  As  suggested  in  § 101,  find  cos  3a;  in  terms  of  cosx. 


104 


PLANE  TRIGONOMETRY 


102.  Functions  of  Half  an  Angle.  If  we  substitute  ^ z for  x in  the 
formulas  cos^  x + sin"*  x = 1 (§  14)  and  cos^  x — sin^  x = cos  2 x (§  101), 
so  as  to  find  the  functions  of  half  an  angle,  we  have 

cos^  ^ z + sin^  ^z  =1, 

and  cos^  i — sin^  i s = cos  z. 

Subtracting,  2 sin’^  ^z  =1—  cos  z ; 


whence 


sill  2 


zt 


N 2 


In  the  above  proof,  if  we  add  instead  of  subtract  we  have 
2 cos^  ^z  =1+  cos  z ; 


whence 


cos 


|z  = ±^ 


+ cosz 


c-  s.  1 sin  4 2!  j cos-4^;  , 

Since  tan  — > and  cot  -^z  = — — — > we  have,  by  dividing, 

COS  ^ sm  ^ 


tan-z  = ±^ 


1 — cos  z 


1 + cos  z 


and 


cot 


cos  z 


cos  z 


These  four  formulas  are  important  and  should  be  memorized. 


From  the  formula  for  tan  4 s can  be  derived  a formula  which  is 
occasionally  used  in  dealing  with  very  small  angles.  In  the  triangle 
ACB  we  have  — 


tan-^  A 


— cos  A 

+ CCS  A 


= ± 


N 


1-^ 

c 


1+5 

c 


= ± 


N 


c — b 
c + 6 


Exercise  45.  Functions  of  Half  an  Angle 

Given  sin  30°  = find  the  values  of  the  following : 

1.  sin  15°.  2.  cos  15°.  3.  tanl5°.  4.  cotl5°.  6.  cot7-|°. 

Given  tan  45°  = 1,  find  the  values  of  the  following : 

6.  sin22.5°.  7.  cos  22.5°.  8.  tan22.5°.  9.  cot22.5°.  10.  cotlli° 

11.  Given  sinx  = 0.2,  find  sin-|-x  and  cos4x. 

12.  Given  cosx  = 0.7,  find  sin  4x,  cos  -^x,  tan  ^x,  and  cot^x. 


SUM  OR  DIFFERENCE  OF  TWO  ANGLES 


105 


103.  Sums  and  Differences  of  Functions.  Since  we  liave  (§§  92,  97) 
sin  (cc  + 1/)  = sin  x cos  y + cos  x sin  y, 
and  sin  (x  — y')=  sin  x cos  y — cos  x sin  y, 

we  find,  by  addition  and  subtraction,  that 

sin  (x  -\-  y)-\-  sin  (x  — y)  — 2 sin  x cos  y, 
and  sin  (x  + y)  — sin  (x  — y)=  2 cos  x sin  y. 

Similarly,  by  using  the  formulas  for  cos  (x  ± y),  we  obtain 
cos  (x  + y)  + cos  (x  — y)z=  2 cos  x cos  y, 
and  cos  (x  y')—  cos  (x  — y)  — — 2 sin  x sin  y. 

By  letting  x y = A,  and  x — y — B,  we  have  x = ^(A  + B),  and 
y = ^(A  — -B),  whence 

sinA  + sinR=  2 sini(A  + R)  cos  |(A  — 5), 

sin  A — sin  R=  2 cos  i(A  + R)  sin  ^(A  — R), 

cosA  + cosR=  2 cos  |(A  + R)  cos  i(A  — R), 

and  cos  A — cos  R = — 2 sin \(A  + R)  sin \(A  — R). 


By  division  we  obtain 
sin  A + sinR 


. , . „ tan  A (A  + R)  cot  (A  — R) : 

sin  A — sin  R ^ J tv  y? 

1 


and  since  cot  ^{A  — B)  — 
we  have 


tan  -^(A  — R) 

sin  A -f  sin  R tan|(A4-R) 


sin  A — sin  R tan  | (A  — R) 

This  is  one  of  the  most  important  formulas  in  the  solution  of  oblique  triangles. 


Exercise  46.  Formulas 


Prove  the  following  formulas: 


1.  sin  2 X 

2.  cos  2 X 


2 tan  a; 

1 + tan^cc 
1 — tan^a; 
1 + tan^a; 


3.  tan^a:  = 

4.  cot^a:  = 


sin  a; 

1 + cos  X 
sin  a: 

1 — cos  X 


If  A,  B,  C are  the  angles  of  a triangle,  prove  that : 

5.  sin  A + sinR  + sin  C = 4 cos  A cos  ^ R cos  ^ C. 

6.  cos  A + cosR  + cosC  = 1 + 4sin4A  sin-|-R  sin -I- (7. 

7.  tan  A + tanR  + tan  C = tan  A tanR  tan  C. 


106 


PLANE  TPIGONOMETKY 


8.  Given  tan  = 1,  find  cos x. 

9.  Given  cot  ^x  = VS,  find  sin x. 

-r,  i.  H oo  sin  33°  + sin  3° 

10.  Prove  that  tan  18  = 


cos  33°  + cos  3° 


.^Lll.  Prove  that  sin-^x  ± cos-|-x  = Vl  ± sinx, 

.r,  , , . tan  X + tan  y 

^ 12.  Prove  that  — ; ; — = + tanx  tan  w. 

cot  X ± cot  y 


13.  Prove  that  tan  (45°  — x)  = 


1 — tan  X 


1 + tan  X 

.^14.  In  the  triangle  ABC  prove  that 

cot  ^ A + cot  j^B  + cot  Y G = cot  cot  j-B  cot  ^ C. 

Change  to  a form  involving  products  instead  of  sums,  and  hence 
more  convenient  for  computation  by  logarithms : 

15.  cot  X + tan  x.  20.  1 + tan  x tan  y. 

16.  cotx  — tanx.  21.  1 — tanx  tany. 

17.  cotx  + tany.  22.  cot  x cot  2/ + 1. 

^18.  cotx  — tan y.  23.  cotxcoty  — 1. 


19. 


1 — COS  2 X 
1 + cos  2 X 


^24. 


tan  X 4-  tan  y 
cot  X + cot  y 


/ 26.  Prove  that  tan  x + tan  y = ^ • 

cos  X cos  y 

-1  , , , sin  (x  — y) 

26.  Prove  that  cot  y — cotx  = — ^ 

sin  X sin  y 

27.  Given  tan  (x  + y)  = 3,  and  tan  x — 2,  find  tan  y. 

28.  Prove  that  (sin  x + cos  xf  = 1 + sin  2 x. 

29.  Prove  that  (sin  x — cos  x)'^  = 1 — sin  2 x. 

30.  Prove  that  tan  x + cot  x = 2 esc  2 x. 

31.  Prove  that  cot  x — tan  x = 2 cos  2 x esc  2 x. 

32.  Prove  that  2 sin^(45°  — x)  = 1 — sin  2 x. 

33.  Prove  that  cos  45°  + cos  75°  = cos  15°. 

34.  Prove  that  1 + tan  x tan  2 x = tan  2 x cot  x — 1. 

Prove  the  folloiving  formulas : 

^35.  (cos  X + cos  yf  + (sin  x + sin  yf  =2  + 2 cos  (x  — y). 

36.  (sin X + cos  yf  + (sin  y + cos  x)^=  2 + 2 sin (x  + y). 

37.  sin  (x  + ?/)  + cos  (x  — y)  = (sin  x + cos  x)  (sin  y + cos  y). 

38.  sin  (x  + y)  cos  y — cos  (x  + y)  sin  y — sin  x. 


CHAPTER  Vn 


THE  OBLIQUE  TRIANGLE 

104.  Geometric  Properties  of  the  Triangle.  In  solving  an  oblique 
triangle  certain  geometric  properties  are  involved  in  addition  to 
those  already  mentioned  in  the  preceding  chapters,  and  these  should 
be  recalled  to  mind  before  undertaking  further  work  with  trigono- 
metric functions.  These  properties  are  as  follows : 

The  angles  opposite  the  equal  sides  of  an  isosceles  triangle 
are  equal. 

If  two  angles  of  a triangle  are  equal,  the  sides  opposite  the  equal 
angles  are  equal. 

If  two  angles  of  a triangle  are  unequal,  the  greater  side  is 
opposite  the  greater  angle. 

If  two  sides  of  a triangle  are  unequal,  the  greater  angle  is 
opposite  the  greater  side. 

A triangle  is  determined,  that  is,  it  is  completely  fixed  in  form 
and  size,  if  the  following  pa7'ts  are  given: 

1.  Two  sides  and  the  included  angle. 

2.  Tivo  angles  and  the  included  side. 

3.  Two  angles  and  the  side  opposite  one  of  them. 

4.  Two  sides  and  the  angle  opposite  one  of  them. 

5.  Three  sides. 

The  fourth  case,  however,  will  be  recalled  as  the  ambiguous  case,  since  the 
triangle  is  not  in  general  completely  determined.  If  we  have  given  and 
sides  a and  h in  this  figure,  either  of  the  triangles  ABC 
&ndAB'C  will  satisfy  the  conditions. 

If  a is  equal  to  the  perpendicular  from  C on  AB,  how- 
ever, the  points  B and  B'  will  coincide,  and  hence  the  two 
triangles  become  congruent  and  the  triangle  is  completely 
determined. 

The  five  cases  relating  to  the  determining  of  a 
triangle  may  be  summarized  as  follows : A triangle  is  determined 
when  three  independent  parts  are  given. 

This  excludes  the  case  of  three  angles,  because  they  are  not  independent. 
That  is,  A = 180°  — {B  + C),  and  therefore  A depends  upon  B and  C. 

107 


108 


PLANE  TPIGONOMETEY 


105.  Law  of  Sines.  In  the  triangle  ABC,  using  either  of  the  figures 
as  here  shown,  we  have  the  following  relations. 


In  the  first  figure, 
and  in  the  second  figure. 


^ • r> 

- = smS, 
a 

- = sin(180° 
a 


B) 


= sin  B. 

Therefore,  whether  h lies  within  or  without  the  triangle,  we 
obtain,  by  division,  the  following  relation : 

a sinA 
b sinE 

In  the  same  way,  by  drawing  perpendiculars  from  the  vertices 
A and  B to  the  opposite  sides,  we  may  obtain  the  following  relations  : 

b sin£ 
c 


and 


sinC 
sin  A 
sin  C 


This  relation  between  the  sides  and  the  sines  of  the  opposite  angles 
is  called  the  Law  of  Sines  and  may  be  expressed  as  follows  : 

The  sides  of  a triangle  are  proportional  to  the  sines  of  the  opposit-e 
angles. 

If  we  multiply  - = by  6,  and  divide  by  sin  A,  we  have 
b sinJ? 

a _ b 
sin  A sin  5 

Similarly,  we  may  obtain  the  following  : 

a _ 6 _ c 

sin  A sinB  sin  C 

and  this  is  frequently  given  as  the  Law  of  Sines. 

It  is  also  apparent  that  a sin  B = b sin  A,  a sin  C = c sin  A,  and  b sin  C = c sin  B, 
three  relations  which  are  still  another  form  of  the  Law  of  Sines. 


THE  OBLIQUE  TKIANGLE 


109 


106.  The  Law  of  Sines  extended.  There  is  an  interesting  extension 
of  the  Law  of  Sines  with  respect  to  the  diameter  of  the  circle  circum- 
scribed about  a triangle. 

Circumscribe  a circle  about  the  triangle  ABC  and  draw  the  radii 
OB,  OC,  as  shown  in  the  figure.  Let  J?  denote  the  radius.  Draw 
OM  perpendicular  to  BC.  Since  the  angle  BOC  is  a central  angle 
intercepting  the  same  arc  as  the  angle  A,  the  angle  BOC  = 2 A-, 
hence  the  angle  BOM  = A\  then 


Therefore 
In  like  manner, 
and 

Therefore 


BM  — R smBOM  = R sin^l. 
a = 2 R sin4. 
b = 2 R sin  B, 
c = 2 R sin  C. 

a b c 


2R  = 


sinG  sin  5 sinC 


A 


That  is,  The  ratio  of  any  side  of  a triangle  to  the  sine  of  the  oppo- 
site angle  is  numerically  equal  to  the  diameter  of  the  circumscribed 
circle. 


Exercise  47.  Law  of  Sines  y 

1.  Consider  the  formula  % — when  B = 90° ; when  A = 90° : 
1 „ 1 , 5 sin  5 ’ 

when  A — B-,  when  a = b. 

^ 2.  Prove  by  the  Law  of  Sines  that  the  bisector  of  an  angle  of  a 
triangle  divides  the  opposite  side  into  parts  proportional  to  the 
adjacent  sides. 

3.  Prove  Ex.  2 for  the  bisector  of  an  exterior  angle  of  a triangle. 

//d.  The  triangle  ABC  has  A = 78°,  B = 72°,  and  c = 4 in.  Find  the 
diameter  of  the  circumscribed  circle. 

^ 5.  The  triangle  ABC  has  A = 76°  37',  B = 81°  46',  and  c = 368.4  ft. 
Find  the  diameter  of  the  circumscribed  circle. 

6.  What  is  the  diameter  of  the  circle  circumscribed  about  an  equi- 
lateral triangle  of  side  7.4  in.  ? What  is  the  diameter  of  the  circle 
inscribed  in  the  same  triangle  ? 

l/  7.  What  is  the  diameter  of  the  circle  circumscribed  about  an  isos- 
celes triangle  of  base  4.8  in.  and  vertical  angle  10°  ? 

\js.  What  is  the  diameter  of  the  circle  circumscribed  about  an  isos- 
celes triangle  whose  vertical  angle  is  18°  and  the  sum  of  the  two  equal 
sides  18  in.  ? 


110 


PLAJSTE  TEIGONOMETRY 


107.  Applications  of  the  Law  of  Sines.  If  we  have  given  any  side 
of  a triangle,  and  any  two  of  the  angles,  we  are  able  to  solve  the  tri- 
angle by  means  of  the  Law  of  Sines.  Thus,  if  we  have  given  a,  A, 
and  B,  in  this  triangle,  we  can  find  the  remaining  parts  as  follows : 


1. 

2. 


3. 


C = 180°  - (A  + B). 


c sin  C a sin  C a 

- — .■.c  = — — — -xsinC. 

a sinA  sinA  sinA 


For  example,  given  a = 24.31,  A = 45°  18',  and  B = 22°  11',  solve 
the  triangle. 

The  work  may  be  arranged  as  follows  : 


a = 24.31 
A = 45°  18' 
B=  22°  11' 
A + E = 67°  29' 
.*.  C = 112°31' 


log  a = 1.38578 
colog  sin  A = 0.14825 
log  sinE  = 9.57700 
logi  = 1.11103 
.-.  b = 12.913 


= 1.38578 
= 0.14825 
log  sin  C = 9.96556 
log  c = 1.49959 
.-.  c = 31.593 


When  — 10  is  omitted  after  a logarithm  or  cologarithm  to  which  it  belongs, 
it  must  still  be  remembered  that  the  logarithm  or  cologarithm  is  10  too  large. 

The  length  of  a having'been  given  only  to  four  significant  figures,  the  values 
of  h and  c are  to  be  depended  upon  only  to  the  same  number  of  significant 
figures  in  practical  measurement.  In  the  above  example  a is  given  to  only  four 
significant  figures,  and  hence  we  say  that  6 = 12.91,  and  c = 31.59. 


Exercise  48.  Law  of  Sines 
Solve  the  triangle  ABC,  given  the  following  parts  : 
. 1.  a = 500,  A = 10°  12',  B = 46°  36'. 

2.  a = 795,  A = 79°  59',  S = 44°  41'. 

3.  a = 804,  A = 99°  55',  B = 45°  1'. 

4.  a = 820,  A = 12°  49',  B = 141°  59'. 

6.  c = 1005,  A = 78°  19',  B = 54°  27'. 

6.  5 = 13.57,  B = 13°  57',  C = 57°  13'. 

7.  a = 6412,  A = 70°  55',  C = 52°  9'. 

Y8.  b = 999,  A = 37°  58',  C = 65°  2'. 


THE  OBLIQUE  TRIANGLE 


111 


Solve  Exs.  9 -14  without  using  logarithms : 

9.  Given  i = 7.071,  A = 30°,  and  C = 105°,  find  a and  c. 

10.  Given  c = 9.562,  A = 45°,  and  B = 60°,  find  a and  b. 

i/ll.  The  base  of  a triangle  is  600  ft.  and  the  angles  at  the  base 
are  30°  and  120°.  Find  the  other  sides  and  the  altitude. 

^l2.  Two  angles  of  a triangle  are  20°  and  40°.  Find  the  ratio  of 

and  the  side  oppo- 

/ site  the  smallest  angle  is  3.  Find  the  other  sides. 

14.  Given  one  side  of  a triangle  27  in.,  and  the  adjacent  angles 
each  equal  to  30°,  find  the  radius  of  the  circumscribed  circle. 

15.  The  angles  B and  C of  a triangle  ABC  are  50°  30'  and  122°  9’ 
respectively,  and  RC  is  9 mi.  Find  AB  and  AC. 

16.  In  a parallelogram,  given  a diagonal  d and  the  angles  x and  y 
which  this  diagonal  makes  with  the  sides,  find  the  sides.  Compute 
the  results  when  d — 11.2,  x = 19°  1',  and  y = 42°  54'. 

17.  A lighthouse  was  observed  from  a ship  to  bear  N.  34°  E.; 
after  the  ship  sailed  due  south  3 mi.  the  lighthouse  bore  N.  23°  E. 
Find  the  distance  from  the  lighthouse  to  the  ship  in  each  position. 

The  phrase  to  hear  N.  34°  E.  means  that  the  line  of  sight  to  the  lighthouse  is 
in  the  northeast  quarter  of  the  horizon  and  makes,  with  a line  due  north,  an 
angle  of  34°. 

/ 18.  A headland  was  observed  from  a ship  to  bear  directly  east ; 

; after  the  ship  had  sailed  5 mi.  N.  31°  E.  the  headland  bore  S.  42°  E. 
FindAhe  distance  from  the  headland  to  the  ship  in  each  position. 

19.  In  a trapezoid,  given  the  parallel  sides  a and  b,  and  the  angles 
X and  y at  the  ends  of  one  of  the  parallel  sides,  find  the  nonparallel 
sides.  Compute  the  results  when  a = 15,  b — 7,  x = 70°,y  = 40°. 

20.  Two  observers  5 mi.  apart  on  a plain,  and  facing  each  other, 
find  that  the  angles  of  elevation  of  a balloon  in  the  same  vertical 

I plane  with  themselves  are  55°  and  58°  respectively.  Find  the  dis- 
tance  from  the  balloon  to  each  observer,  and  also  the  height  of  the 
balloon  above  the  plain. 

21.  A balloon  is  directly  above  a straight  road  7^  mi.  long,  joining 
two  towns.  The  balloonist  observes  that  the  first  town  makes  an 
angle  of  42°  and  the  second  town  an  angle  of  38°  with  the  perpen- 
dicular. Find  the  distance  from  the  balloon  to  each  townj  and  also 
the  height  of  the  balloon  above  the  plain. 

/ ^ . 


the  opposite  sides,  v/ 

131  'i'he^  angles  of  a triangle  are  as  5:10:  21, 


112 


PLANE  TRIGONOMETRY 


108.  The  Ambiguous  Case.  As  mentioned  in  § 104,  if  two  sides 
of  a triangle  and  the  angle  opposite  one  of  them  are  given,  the  solu- 
tion will  lead,  in  general,  to  two  triangles.  Thus,  if  we  have  the 
two  sides  a and  b and  the  angle  A given,  we  proceed  to  solve  the 
triangle  as  follows : 

C = 180°  - (d  + B)  ; 

hence  we  can  find  C if  we  can  find  B. 

„ c sin  C 

I urthermore,  - = ? 

a sin  A 

, a sin  C 

whence  c = — : — — ; 

sm  A 

hence  we  can  find  c if  we  can  find  C,  and  we  can  also  find  c if  we 
can  find  B.  But  to  find  B we  have 

sin  B _h 

sin  A a 

, . Zisind 

whence  sin  B = — • 

a 


Therefore  we  do  not  find  B directly,  but  only  sin  B.  But  when  an 
angle  is  determined  by  its  sine,  it  admits  of  two  values  which  are 
supplements  of  each  other  (§  86) ; hence  either  of  the  two  values 
of  B may  be  taken  unless  one  of  them  is  excluded  by  the  conditions 
of  the  problem. 

In  general,  therefore,  either  of  the  triangles  ABC  and  AB'C  fulfills 
the  given  conditions. 


Exercise  49.  The  Ambiguous  Case 


In  the  triangle  AB  C given  a,  b,  and  A,  prove  that : 

1.  If  a > h,  then  d > R,  A is  acute,  and  there  is  one  and  only  one 
triangle  which  will  satisfy  the  given  conditions. 

2.  If  a = h,  both  A and  B are  acute,  and  there  is  one  and  only  one 
triangle  which  will  satisfy  the  given  conditions,  and  this  triangle  is 
isosceles. 

3.  If  a < h,  then  d must  be  acute  to  have  the  triangle  possible,  and 
there  are  in  general  two  triangles  which  satisfy  the  given  conditions. 

4.  If  a = 6 sind,  the  required  triangle  is  a right  triangle. 

6.  If  a.<Z»sind,  the  triangle  is  impossible. 

6.  If  d = B,  there  is  one,  and  only  one,  triangle. 


THE  OBLIQUE  TKIANGLE 


113 


109.  Number  of  Solutions  to  be  expected.  We  may  summarize  th.6 
results  found  on  page  112  as  follows : 

There  are  two  solutions  if  A is  acute  and  the  value  of  a lies  be- 
tween h and  h sin  A. 

There  is  no  solution  if  A is  acute  and  a<.b  sin  A ; or  if  A is  obtuse 
and  a<.b,  or  a = b. 

There  is  one  solution  in  each  of  the  other  cases. 

The  number  of  solutions  can  often  be  determined  by  inspection.  In  case  of 
doubt,  find  the  value  of  h sin^. 

We  can  also  determine  the  number  of  solutions  by  considering  the  value  of 
log  sin  B.  If  log  sin  5 = 0,  then  sin  5 = 1 and  5 = 90°.  Therefore  the  triangle 
required  is  a right  triangle.  If  log  sin  5 > 0,  then  sin  5 > 1,  and  hence  the 
triangle  is  impossible.  If  log  sin  5 < 0,  there  is  one  solution  when  a>b  ; there 
are  two  solutions  when  a <h. 

When  there  are  two  solutions,  let  5',  C\  c',  denote  the  unknown  parts  of  the 
second  triangle  ; then 

5' = 180° -5, 

C'  = 180°-  {A-+B')  = B-A, 

, , a sin  C' 

and  c = 

sin^ 

110.  Illustrative  Problems.  The  following  may  be  taken  as  illus- 
trative of  the  above  cases  : 

1.  Given  a = 16,  b = 20,  and  A = 106®,  find  the  remaining  parts. 

In  this  case  a <b  and  A > 90°.  Since  a<b,  it  follows  that  A<B.  Hence  if 

A > 90°,  5 must  also  be  greater  than  90°.  But  a triangle  cannot  have  two 
obtuse  angles.  Therefore  the  triangle  is  impossible. 

2.  Given  a = 36,  b 80,  and  A = 30°,  find  the  remaining  parts. 
Here  we  have  hsinH  = 80  x | = 40 ; so  that  a < hsinJ.  and  the  triangle  is 

impossible.  Draw  the  figure  to  illustrate  this  fact. 

3.  Given  a = 25,  b = 50,  and  A = 30°,  find  the  remaining  parts. 
Here  we  have  b sin  ^ = 60  x ^ = 25  ; but  a is  also  equal  to  25.  Hence  5 

must  be  a right  angle.  ABC  is  therefore  a right  triangle  and  there  is  only  one 
solution. 

4.  Given  a = 30,  b = 30,  and  A = 60°,  find  the  remaining  parts. 
Here  we  have  a = b,  and  A an  acute  angle.  Hence  there  is  one  solution  and 

only  one.  It  is  evident,  also,  that  the  triangle  is  not  only  isosceles  but  equilateral. 

6.  Given  a = 3.4,  b = 3.4,  and  A = 45°,  find  the  remaining  parts. 
Here  we  have  a = b,  and  A an  acute  angle.  Hence  there  is  one  solution  and 
only  one.  It  is  evident,  also,  that  the  triangle  is  not  only  isosceles  but  right. 


114 


PLANE  TKIGONOMETRY 


6.  Given  a = 72,630,  h = 117,480,  and  A = 80°  0'  50",  find  B, 
C,  and  c. 

log  h = 5.06997  Here  log  sin  5 > 0. 
log  sin  A = 9.99337  Therefore  sin  iJ>l,  which  is  impossible. 

colog  a = 5.13888 
log  sin  B = 0.20222 
Therefore  there  is  no  solution. 


7.  Given  a = 13.2,  h = 15.7,  and  A = 57°  13'  15",  find  B,  C,  and  c. 


logs  = 1.19590 
log  sin  A = 9.92467 
colog  a = 8.87943 
log  sin  B = 0.00000 
J5  = 90° 

C = 32°  46' 45" 


c = h cos  A 
log  h = 1.19590 
log  cos  A = 9.73352 
logc  = 0.92942 
c = 8.5 


Therefore  there  is  one  solution. 

Since  B = 90°,  the  triangle  is  a right  triangle. 


8.  Given  a = 767,  S = 242,  and  A = 36°  53'  2",  find  B,  C,  and  c. 


logs  = 2.38382 
log  sin  A = 9.77830 
colog  a ■ 7.11520 
log  sin  B = 9.27732 
.-.5=  10°  54' 58" 
.-.  C=132°12'0" 


log  a = 2.88480 
log  sin  C = 9.86970 
colog  sin  A = 0.22170 
logc  = 2.97620 
c = 946.68 
= 946.7 


Here  a > S,  and  log  sin  .B  < 0. 
Therefore  there  is  one  solution. 


9.  Given  a = 177.01,  S = 216.45,  and  A = 35°  36'  20",  find  the 
other  parts. 


logs  = 2.33536 
log  sin  A = 9.76507 
colog  a = 7.75200 
log  sin  B = 9.85243 

A = 45°  23' 28"  or 
134°  36'  32" 
e=99°0'12"or 
9°  47' 8" 


log  a = 2.24800 
log  sin  C = 9.99462 
colog  sin  A = 0.23493 
logc  = 2.47755 


2.24800 

9.23035 

0.23493 

1.71328 


c = 300.29  or  51.675 
= 300.29  or  51.68 


Here  a < b,  and  log  sin  B < 0. 
Therefore  there  are  two  solutions. 


THE  OBLIQUE  TEIANGLE 


115 


Exercise  50.  The  Oblique  Triangle 

Find  the  number  of  solutions,  given  the  following : 


1.  a = 80, 

b = 100, 

A = 30°. 

2.  a = 50, 

b = 100, 

A = 30°. 

3.  a = 40, 

b = 100, 

A = 30°. 

4.  (z  = 100, 

b = 100, 

A = 30°. 

5.  a = 13.4, 

b = 11.46, 

A = 77°  20'. 

6.  a = 70, 

b = 75. 

A = 60°. 

7.  a = 134.16, 

b = 84.54, 

B = 52°  9'. 

8.  a = 200, 

b = 100, 

A = 30°. 

the  triangles,  given  the  following : 

9.  a = 840, 

b = 485, 

A = 21°  31'. 

10.  a = 9.399, 

b = 9.197, 

A = 120°  35'. 

11.  a = 91.06, 

b = 77.04, 

A = 51°  9'. 

12.  a = 55.55, 

b = 66.66, 

B = 77°  44'. 

13.  a = 309, 

b = 360, 

A = 21°  14'. 

14.  a = 34, 

b = 22, 

B = 30°  20'. 

15.  b = 19, 

c = 18. 

C=15°  49'. 

16.  a = 8.716, 

b = 9.787, 

A = 38°  14'  12' 

17.  a = 4.4, 

b = 5.21, 

A =57°  37' 17 

18.  Given  a = 75,  b = 29,  and  B = 16°  15',  find  the  difference  be- 
tween the  areas  of  the  two  triangles  which  meet  these  conditions. 

19.  In  a parallelogram,  given  the  side  a,  a diagonal  d,  and  the 
angle  A made  by  the  two  diagonals,  find  the  other  diagonal.  As  a 
special  case  consider  the  parallelogram  in  which  a = 35,  d = 63, 
and  A = 21°  36'. 

20.  In  a parallelogram  ABCD,  given  AD  = 3 in.,  BD  = 2.5  in.,  and 
A = 47°  20',  find  AB. 

21.  In  a quadrilateral  ABCD,  given  AC  = 4 in.,  /LB AC  = 35°, 
AB  = 75°  20',  Ad  = 38°  30',  and  ABAD  = 70°  40',  find  the  length 
of  each  of  the  four  sides. 

22.  In  a pentagon  ABCDE,  given  AA  = 110°  50',  AB  = 106°  30', 
ZE  = 104°10',  ABAC  = 3i)°,  ADAE  = 34.°  5&,  AADC  = 52°  30', 
and  AC  = 6 in.,  find  the  sides  and  the  remaining  angles  of  the 
pentagon. 


116 


PLANE  TPIGONOMETEY 


111.  Law  of  Cosines.  This  law  gives  the  value  of  one  side  of 
a triangle  in  terms  of  the  other  two  sides  and  the  angle  included 
between  them. 


In  either  figure,  + BD-. 

In  the  first  figure,  B'  —c—AD. 

In  the  second  figure,  BD  =AD  — c. 

In  either  case,  BD^  =AD^  — 2 c x AD  + 

Therefore,  in  all  cases,  -\-  AD^  — 2 c x AD. 

Now  K^-\-AD^  — V^, 

and  AD  = b cos^l. 

Therefore  a*  = 6^  + — 2 &c  cos  A. 

In  like  manner  it  may  be  proved  that 

b“^  — + a?  — 2 ca  cosB, 

and  (^  = + b^  — 2 ab  cos  C. 


The  three  formulas  have  precisely  the  same  form,  and  the  Law 
of  Cosines  may  be  stated  as  follows : 

The  square  on  any  side  of  a triangle  is  equal  to  the  sum  of  the 
squares  on  the  other  two  sides  diminished  hy  twice  their  product  into 
the  cosine  of  the  included  angle. 


It  will  be  seen  that  if  A = 90°,  we  have 

= 6-  + — 2 6c  cos  90° 

= 62  + c2. 

In  other  words  we  have  the  Pythagorean  Theorem  as  a special  case.  Hence 
this  is  sometimes  called  the  Generalized  Pythagorean  Theorem. 

It  will  also  be  seen  that  the  law  includes  two  other  familiar  propositions  of 
geometry,  one  of  which  is  the  following : 

In  an  obtuse  triangle  the  square  on  the  side  opposite  the  obtuse  angle  is  equivalent 
to  the  sum  of  the  squares  on  the  other  two  sides  increased  by  twice  the  product  of 
one  of  those  sides  by  the  projection  of  the  other  upon  that  side. 

This  and  the  analogous  proposition  are  given  as  exercises  on  page  117. 


THE  OBLIQUE  TRIANGLE 


117 


Exercise  51.  Law  of  Cosines 


1.  Using  the  figures  on  page  116,  prove  that,  whether  the  angle 
B is  acute  or  obtuse,  c = a cos  B -\-h  cos^l. 

2.  What  are  the  two  symmetrical  formulas  obtained  by  changing 
the  letters  in  Ex.  1 ? What  does  the  formula  in  Ex.  1 become  when 
R = 90°  ? 


3.  Show  that  the  sum  of  the  squares  on  the  sides  of  a triangle 
is  equal  to  2(ab  cos  C + be  cos  J + ca  cosR). 

4.  Consider  the  Law  of  Cosines  in  the  case  of  the  triangle  a = 5, 
b = 12,  c = 6. 

5.  Given  a = 5,  b = 5,  and  C = 60°,  find  c. 

6.  Given  a ~ 10,  b — 10,  and  C = 45°,  find  c. 

7.  Given  a = S,  b = 5,  and  C — 60°,  find  c. 

8.  From  the  formula  = b^  + — 2 be  cos  A deduce  a formula 
for  cosd.  From  this  result  find  the  value  of  A when  b^  z=  a^. 


9.  Prove  that  if 
right. 


cos  A cos  B 


a 


the  triangle  is  either  isosceles  or 


^ cos  A , cos  B cos  C 

10.  Prove  that  1 ; 1 = — ^ -- — 

a b c 2 abc 


11.  Prove  that  — cos  A + t cos  B + — cos  C = 
abc 


2 abc 


12.  From  the  Law  of  Cosines  prove  that  the  square  on  the  side  op- 
posite an  acute  angle  of  a triangle  is  equal  to  the  sum  of  the  squares 
on  the  other  two  sides  minus  twice  the  product  of  either  side  and 
the  projection  of  the  other  side  upon  it. 

13.  As  in  Ex.  12,  consider  the  geometric  proposition  relating  to 
the  square  on  the  side  opposite  an  obtuse  angle. 

14.  In  the  parallelogram  ABCD,  given  AB  = 4 in.,  AD  = 5 in.,  and 
A = 38°  40',  find  the  two  diagonals. 

15.  In  the  parallelogram  ABCD,  given  AB  = 7 in.,  AC  = 10  in., 
and  /.BAC  = 36°  7',  find  the  side  BC  and  the  diagonal  BD. 

16.  In  the  quadrilateral  ABCD,  given  AC  = 3.6  in.,  AZ)  = 4 in., 
BC  = 2.4  in.,  Z.ACB  = 29°  40',  and  Z. CAD  = 71°  20',  find  the  other 
two  sides  and  all  four  angles  of  the  quadrilateral. 

17.  In  the  pentagon  ABCDE,  given  AB=  3.4  in.,  AC  = 4.1  in., 
AZ>  = 3.9in.,  AE=2.2in.,  ZBAC  = ?,^°V,  Z CAD  = 41°  22',  and 
ZDAE  = 32°  5',  find  the  perimeter  of  the  pentagon. 


118 


PLANE  TRIGONOMETRY 


iin  -r  , ^ 0(-  ® sin  4 

112.  Law  of  Tangents.  Since  t = ^ > 

^ b sinR 

follows  by  the  theory  of  proportion  that 


by  the  Law  of  Sines,  it 


a — b _ sin  4 — sinR 
a b sinA  + sinR 


This  is  easily  seen  without  resorting  to  the  theory  of  proportion.  For,  since 
a sin  J5  = 6 sin  A (§  105),  we  have 

a sinil  — 6 sinA  = 6 sinA—  a sinB 
Adding,  a sin  A — 6 sin  i?  = a sin  A—h  sin  B 

a sinA  + asinB  — hsin  A — 6sinJ5  = a sinA  — a sin5  + 6 sin  A — 6 sin  5, 

or  (a  — h)  (sin  A + sin  B)  = {a  + 6)  (sin  A — sin  B) ; 

, , ....  a — b sin  A — sin  B 

whence,  by  division,  

a + 6 sin  A + sin  B 


But  by  § 103, 
Therefore 


sinA  — sinR  _ tan  1 (A  — B) 
sin  A + sin  R tan  ^ (A  + R) 

a — b tan  |(A  — R) 
a-\-  b tan |(A  + R) 


By  merely  changing  the  letters. 


a — c _ tan  ^ (A  — C) 
a + c tan  ^ (A  + O)  ’ 

and  ^^tan^(R-C)^ 

b + c tan^(R+C) 

Hence  the  Law  of  Tangents  : 

The  difference  between  two  sides  of  a triangle  is  to  their  sum  as 
the  tangent  of  half  the  difference  between  the  ojojoosite  angles  is  to 
the  tangent  of  half  their  sum. 

In  the  case  of  a triangle,  if  we  know  the  two  sides  a and  b and 
the  included  angle  C,  we  have  our  choice  of  two  methods  of  solving. 
From  the  Law  of  Cosines  we  can  find  c,  and  then,  from  the  Law  of 
Sines,  we  can  find  A and  R.  Or  we  can  find  A B by  taking  C from 
180°,  and  then,  since  we  also  know  a-\-b  and  a — b,  we  can  find 
A — B.  From  A + R and  A — R we  can  find  A and  R.  This  second 
method  is  usually  the  simpler  one. 

If  6 > a,  then  R > A . The  formula  is  still  true,  but  to  avoid  negative  numbers 
the  formula  in  this  case  should  be  written 

b — a _ tan  ^ (R  — A) 

6 + a tan  ^ (R  + A) 


THE  OBLIQUE  TRIANGLE 


119 


Exercise  52.  Law  of  Tangents 


Find  the  form  to  which 

a^h 


tan  1(A  — B) 
tan  1 (A  + B) 


reduces  when : 


1.  C=  90°.  3.  A=B^C. 

2.  a = b.  A.  A -B  = 90°,  and  B = C, 


Prove  the  following  formulas  ; 
b - c 


7. 


8. 


9. 


10. 


= tan^(R-C)cot^(R  + C'). 
b — c 


■ cot  ^A. 


b + c 

tan  \(B~C)  , 

^ ^ ^ b c 

a -\-b  _ cot  ^{A~B) 

a — b cot  -^  ( A + A) 

sin  A 4-  sinR  _ tan  ^ (A  + B) 

sin  A — sin  R tan-|-(A— R) 

sinR  + sin  C _ 2 sin  (R  + C)  cos  ^(B  — C) 

sin  R — sin  C 2 cos  ^ (R  + C)  sin  (R  — C) 

sin  A + sinR 

sin  A — sinR 


= tan  ^ (A  + R)  cot  ^ (A  — R). 


11.  To  what  does  the  formula  in  Ex.  8 reduce  when  A = R ? 

12.  To  what  does  the  formula  in  Ex.  9 reduce  when  B — C = 60°  ? 

13.  To  what  does  the  formula  in  Ex.  10  reduce  when  the  triangle 
is  equilateral  ? 

14.  To  what  does  the  Law  of  Tangents,  in  the  form  stated  at  the 
top  of  this  page,  reduce  in  the  case  of  an  isosceles  triangle  in  which 
a = b?  What  does  this  prove  with  respect  to  the  angles  opposite 
the  equal  sides  ? 

15.  By  the  help  of  the  Law  of  Tangents  prove  that  an  equilateral 
triangle  is  also  equiangular. 

16.  By  the  help  of  the  Law  of  Tangents  prove  that  an  equiangular 
triangle  is  also  equilateral.  . 

17.  Given  any  three  sides  and  any  three  angles  of  a quadrilateral, 
show  how  the  fourth  side  and  the  fourth  angle  can  be  found.  Show 
also  that  it  is  not  necessary  to  have  so  many  parts  given,  and  find 
the  smallest  number  of  parts  that  will  solve  the  quadrilateral. 

18.  What  sides,  what  diagonals,  and  what  angles  of  a pentagon  is  it 
necessary  to  know  in  order,  by  the  aid  of  the  Law  of  Tangents  alone, 
to  solve  the  pentagon  ? 


120 


PLANE  TPIGONOMETKY 


113.  Applications  to  Triangles.  The  Law  of  Cosines  and  the  Law 
of  Tangents  are  frequently  used  in  the  solution  of  triangles.  This 
is  particularly  the  case  when  we  have  given  two  sides,  a and  h,  and 
the  included  angle  C. 

There  are  two  convenient  ways  of 
finding  the  angles  A and  B,  the  first  being 
by  the  Law  of  Tangents.  This  law  may 
be  written 

tan  ^(A—B)=  X tan  J (A  + A). 

Since  -^(A  + 5)  = ^(180°—  C),  the  value  of  ^(A  + A)  is  known,  so 
that  this  equation  enables  us  to  find  the  value  of  ^(A—B).  We 
then  have  ^(^a+B)  + i(A—B)  = A, 

and  ^(A+B)  — ^(A—B)=B. 

The  second  method  of  finding  A and  B is  as  follows  : In  the  above 
figure  let  BD  be  perpendicular  to  A C. 

BD  BD 


Then 

Now 

and 


tanA  = - — = 


AD  AC— DC 
BD  = a sin  C, 
DC  = a cos  C. 

a sin  C 


tanA 


b — a cos  C 


Since  A and  C are  now  known,  B can  be  found. 

This  is  not  so  convenient  as  the  first  method,  because  it  is  not  so  well  adapted 
to  work  with  logarithms. 

The  side  c may  now  be  found  by  the  Law  of  Sines,  thus : 

a sin  C b sin  C 

c = — : — — j or  c = — ^ — — • 
sin  A sin  A 


Instead  of  finding  A and  B first,  and  from  these  values  finding  c, 
we  may  first  find  c and  then  find  A and  B.  To  find  c first  we  may 
write  the  Law  of  Cosines  (§  111)  as  follows  : 

c = ^ o?  b‘‘‘  — 2 oib  cos  C. 

Having  thus  found  c,  and  already  knowing  a,  6,  and  C,  we  have 

a sin  C . b sin  C 

sin  A = > sin5  = 

c c 

In  general  this  is  not  so  convenient  as  the  first  method  given  above,  because 
the  formula  for  c is  not  so  well  adapted  to  work  with  logarithms. 


THE  OBLIQUE  TKIAHGLE 


121 


114.  Illustrative  Problems.  1.  Given  C = 63°  35'  30",  a =748,  and 
b = 375,  find  A,  B,  and  c. 

We  see  that  a.  + 5 = 1123,  a~h—  373,  and  A -\-B  = 180°  — C = 
116°  24'  30".  Hence  ^(A  +B)=5S°  12'  15". 


log  (a  — b')—  2.57171 
colog  (a  + Z»)  = 6.94962 
log  tan  ^ (G  -}-  i?)  z=  0.207 66 
log  tan  ^ (A  —B)=  9.72899 
.-.  ^(A-B)=28°  10'  54" 


log  5 = 2.57403 
log  sinC  = 9.95214 
colog  sin  B = 0.30073 
log  c = 2.82690 
.-.  c = 671.27 


After  finding  —B)  we  combine  this  with  j-(A  +B)  and  find 
A = 86°  23'  9"  and  B = 30°  1'  21". 

In  the  above  example,  in  finding  the  side  c we  use  the  angle  B rather  than 
the  angle  A,  because  A is  near  90°.  The  use  of  the  sine  of  an  angle  near  90° 
should  be  avoided,  because  it  varies  so  slowly  that  we  cannot  determine  the 
angle  accurately  when  the  sine  is  given. 


2.  Given  a.  = 4,  c = 6,  and  B = 60°,  find  the  third  side  b. 

Here  the  Law  of  Cosines  may  be  used  to  advantage,  because  the  numbers 
are  so  small  as  to  make  the  computation  easy.  We  have 

b = Va^  + — 2 ac  cos B = Vl6  + 36  — 24  = ; 

log  28  = 1.44716,  log  = 0.72358,  = 5.2915  ; 

that  is,  to  three  significant  figures,  b = 5.292. 


Exercise  53.  Solving  Triangles 

Solve  these  triangles,  given  the  folloiving  parts : 


1. 

II 

b = 83.39, 

C = 72°  15'. 

2. 

b = 872.5, 

c = 632.7, 

A = 80°. 

3. 

a =17, 

II 

C = 59°  17'. 

4. 

II 

c = V3, 

A = 35°  53'. 

6. 

a = 0.917, 

b = 0.312, 

C = 33°  7'  9". 

6. 

a =13.715, 

c =11.214, 

A =15°  22' 36". 

7. 

b = 3000.9, 

c=  1587.2, 

A = 86°  4'  4". 

8. 

a - 4527, 

b - 3465, 

C = 66°  6'  27". 

9. 

a = 55.14, 

b = 33.09, 

C = 30°  24'. 

10. 

a = 47.99, 

b = 33.14, 

C=175°  19' 10". 

11. 

a = 210, 

b =105, 

C = 36°  52'  12". 

12. 

a = 100, 

b = 900, 

C = 65°. 

122 


PLANE  TKIGONOMETEY 


Solve  these  triangles,  given  the  following  parts : 

13.  a = 409,  &=169,  C=  117.7°. 

14.  a = 6.25,  5 = 5.05,  C=  105.77°. 

15.  a = 3718,  5 =1507,  C = 95.86°. 

16.  a = 46.07,  h = 22.29,  C = 66.36°. 

17.  h = 445,  c = 624,  A = 10.88°. 

18.  b = 15.7,  c = 43.6,  A = 57.22°. 


19.  If  two  sides  of  a triangle  are  each,  equal  to  6,  and  the  in- 
cluded angle  is  60°,  find  the  third  side  by  two  different  methods. 

20.  If  two  sides  of  a triangle  are  each  equal  to  6,  and  the  in- 
cluded angle  is  120°,  find  the  third  side  by  three  different  methods. 

21.  Apply  the  first  method  given  on  page  120"to  the  case  in  which 
a is  equal  to  b ; that  is,  the  case  in  which  the  triangle  is  isosceles. 


22.  If  two  sides  of  a triangle  are  10  and  11,  and  the  included 
angle  is  50°,  find  the  third  side. 

23.  If  two  sides  of  a triangle  are  43.301  and  25,  and  the  included 
angle  is  30°,  find  the  third  side. 

r 24.  In  order  to  find  the  distance  between  two  objects,  A and  B, 
separated  by  a swamp,  a station  C was  chosen,  and  the  distances 
CA  = 3825  yd.,  CB  = 3475.6  yd.,  together  with 
the.^ngle  ACB  = 62°  31',  were  measured.  Eind 
distance  from  to  P. 

\/  25.  Two  inaccessible  objects,  A and  B,  are 
each  viewed  from  two  stations,  C and  B,  on  the 
same  side  of  AB  and  562  yd.  apart.  The  angle 
ACB  is  62°  12',  PCZ»  41°  8',  DP  60°  49',  and 
^BC  34°  51'.  Eequired  the  distance  AB. 

26.  In  order  to  find  the  distance  between  two  objects,  A and  B, 
separated  by  a pond,  a station  C was  chosen,  and  it  was  found  that 
CA  = 426  yd.,  CB  = 322.4  yd.,  and  ACB  = 68°  42'.  Eequired  the 
distance  from  A to  B. 


i 


27.  Two  trains  start  at  the  same  time  from  the  same  station  and 
move  along  straight  tracks  that  form  an  angle  of  30°,  one  train  at 
the  rate  of  30  mi.  an  hour,  the  other  at  the  rate  of  40  rui.  an  hour. 
How  far  apart  are  the  trains  at  the  end  of  half  an  hour  ? 

28.  In  a parallelogram,  given  the  two  diagonals  5 and  6 and  the 
angle  which  they  form  49°  18',  find  the  sides. 


THE  OBLIQUE  TKIAHGLE 


123 


115.  Given  the  Three  Sides.  Given  the  three  sides  of  a triangle,  it 
is  possible  to  find  the  angles  by  the  Law  of  Cosines.  Thus,  from 

a?  z=  — 2 be  cos 4, 

~ a? 


we  have 


cosyI  = 


2 be 


This  formula  is  not,  however,  adapted  to  work  with  logarithms.  In  order  to 
remedy  this  difficulty  we  shall  now  proceed  to  change  its  form. 

Let  s equal  the  semiperimeter  of  the  triangle ; that  is. 


let 

Then 

and 

Hence 


a-\-b-\-e  — 2 s. 

b e — a = 2 s — 2 a = 2 (s  — a), 
e a — b = 2(s  — b), 

a b — e = 2 (s  — e'). 


1 - cos  .4  = 1 — 


b‘^  + e^ 


a 


2be-b^-(^  + a^ 


a 


2 be  2 be 

^ - -(b  — eY  _ (a  b — e){a  — b e) 
2 be  2be 


_ 2(s-b){s-e) 
be 

In  the  same  way  the  value  of  1 + cos  .4  is 


1 + 


b"^  +<?  ~ _ 2 be  Ir  _ (b  cf  — 


2 be 


2 be 


2 he 


(5  + c + g)  (6  + c — a)  _ 2 5 (s  — a) 
2 be  be 

But  from  § 102  we  know  that 

1 — cosh.  = 2 sin^^.4,  and  1 + cosh  = 2 cos^^h. 

. r.  • - 2 (s  — 5)  (s  — c)  , ^ , , , 2 s (s  — a) 

. . 2 simlh  = — ^ -j  and  2 cos^l-h  = — 

^ be  ^ be 

It  therefore  follows  that 

sin  1 h = 


N 


(5_&)(S_C) 


he 


and 


cos  ih. 


Furthermore,  since  tan  x 


sin  X 

! 

COS  X 


s(s  — a) 


be 


we  have 


tan 


|(s-&)(s-c) 
2^  y s(s-a) 


124 


PLANE  TRIGONOMETRY 


By  merely  changing  the  letters  in  the  formulas  given  on  page  123, 
we  have  the  following : 


sin  = 
cos  ^ R = 


N 


(s  - a)  (s  - c)  ^ 


ac 


tan  ^ B 


-4 


(s  — a)  (s  — c) 
s(s  — b) 


sin  A C = -y 

1 

1 

ab 

cos  A C*  = 

s(s  — c) 
ab  ’ 

tan  A C = ^ 

(s  - a)  (s-  b) 

s(s  — c) 

There  is  then  a choice  of  three  different  formulas  for  finding  the  value  of 
each  angle.  If  half  the  angle  is  very  near  0°,  the  formula  for  the  cosine  will 
not  give  a very  accurate  result,  because  the  cosines  of  angles  near  0°  differ  little 
in  value ; and  the  same  is  true  of  the  formula  for  the  sine  when  half  the  angle 
is  very  near  90°.  Hence  in  the  first  case  the  formula  for  the  sine,  and  in  the 
second  that  for  the  cosine,  should  be  used. 

But  in  general  the  formulas  for  the  tangent  are  to  be  preferred,  the  tangent 
as  a rule  changing  more  rapidly  than  the  sine  or  cosine. 

It  is  not  necessary  to  compute  by  the  formulas  more  than  two  angles,  for 
the  third  may  then  be  found  from  the  equation  A + B + C = 180°.  There  is  this 
advantage,  however,  in  computing  all  three  angles  by  the  formulas,  that  we 
may  then  use  the  sum  of  the  angles  as  a test  of  the  accuracy  of  the  results. 


116.  Checks  on  the  Angles.  In  case  it  is  desired  to  compute  all  the 
angles  for  the  purpose  of  checking  the  work,  the  formulas  for  the 
tangent  may  be  put  in  a more  convenient  form. 

The  formula  for  tan  may  be  written  thus  : 


tan  AA  = ^ 

1 

1 

1 

s(s  — ay 

1 

(s  — a)  (s  — b)(s  — c) 

5 

s 

Hence,  if  we  put  ^ = \ 

(s  — a)  (s  — b)  (s  — c) 

we  have  taniA  = 

s s — a 


Likewise, 


tan  = 


tan  A C = 

s — c 


For  example,  if  a = 3,  6 = 3.5,  and  c = 4.5,  we  have  s = 5.5,  s — a = 2.5. 
s — b = 2,  and  s — c = 1. 

...  r = . MZlZi  = .II.=  y!^=  0.9534. 

\ 5.5  \5.5  \11 


...  tan  A A = 0.9534  -h  2.5  = 0.3814. 
...  }A  = 20°  53'. 

...  A = 41°  46'. 


THE  OBLIQUE  TRIANGLE 


125 


Exercise  54.  Formulas  of  the  Triangle 
1.  Given  tan  i ^4  = 


(s-b)(s-c) 


s(s  — a) 


express  the  value  of  log  tan  ^A. 


2.  Given  sin  ^A  = -yA- ^ j express  the  value  of  log  sin  \A. 

^ (5  — 5)  (s  — c)  , , , „ , 

3.  Given  r = j express  the  value  ot  log?’. 


4.  Given  tan ^A  = 


s — a 


5.  Given  tan  ^A  = 


s — a 

6.  Of  the  three  values  for  tan-^^, 


express  the  value  of  log  tan  ^^4. 


express  the  value  of  log  r. 


4 


— cos^ 
+ cos  A ‘ 


(s  -b)(s-c) 
s(s  —■  a) 


and 


— a)  (s  — b)  (s  — c) 


(§  102) 
(§  115) 
(§  116) 


which  is  the  easiest  to  treat  by  logarithms  ? Express  the  logarithms 
of  the  results  and  show  why  your  answer  is  correct. 

7.  Given  a = 4,  b = 5,  and  c = 6,  find  the  value  of  tan  ^A,  and 
then  find  the  value  of  A. 

8.  Deduce  the  equation 


tan  ^ A 


5 -b)(s-  c) 
s(s  — a) 


from  the  equation 


tan  5 ^ 


1 — cos  .4 


1 + cos  A 


9.  Discuss  the  formula 


tan^^ 


^-4 


(s  -b)(s-  c) 
s(s  — a) 

1 

s — a A 


(s  — a)  (s  — b)(s  — c) 


for  the  case  of  an  equilateral  triangle,  say  when  a = 4. 


126 


PLANE  TEIGONOMETPY 


117.  Illustrative  Problems.  1.  Given  a ~ 3.41,  h = 2.60,  c = 1.58, 
find  the  angles. 

Since  it  is  given  that  a = 3.41,  & = 2.60,  and  c = 1.58,  it  follows 
that  2 s = 7.59  and  s = 3.795.  Therefore 

s - a = 0.385,  s-h=  1.195,  s - c = 2.215. 


Using  the  formula  of  § 115  and  the  corresponding  formula  for 
tan  ^B,  we  may  arrange  the  work  as  follows  : 


cologs  = 9.42079 
colog  (s  — a)  = 0.41454 
log  (s  - ^<)=  0.07737 
log(s  - c)=  0.34537 
2)0.25807 
log  tan  = 0.12903 
.-.  JA=  53°  23' 20" 
A = 106°  46'  40" 


cologs=  9.42079  —10 
log  (s  — a)  = 9.58546  — 10 
colog  (s- 6)  = 9.92263-10 
log  (s  — c)  = 0.34537 

2)19.27425  -20 
logtan^£=  9.63713-10 
J.B  = 23°  26' 37" 

.-.  A = 46°  53' 14" 


A -\-B  = 153°  39'  54",  and  C = 26°  20'  6". 

2.  Solve  the  above  problem  by  finding  all  three  angles  by  the  use 
of  the  formulas  on  page  124. 

Since  it  is  given  that  a =3.41,  6=2.60,  and  c = 1.58,  it  follows 
that  2s  = 7.59  and  s = 3.795.  Therefore 


s - a = 0.385,  s - 6 = 1.195,  s - c = 2.215. 


Here  the  work  may  be  compactly  arranged  as  follows,  if  we  find  log  tan A, 
etc.,  by  subtracting  log  (s  — a),  etc.,  from  log  r instead  of  adding  the  cologarithm. 


log(s  — a)  = 9.58546 
log  (s  — 6)  = 0.07737 
log  (s-e)=  0.34537 
cologs  = 9.42079 
log  r"=  9.42899 
logr  = 9.71450 


log  tan  ^A  =10.12903 
logtan4A=  9.63713 
log  tan  i C = 9.36912 


i.4  = 

53° 

23' 

20" 

45  = 

23° 

26' 

37" 

4(7  = 

13° 

10' 

3" 

A = 

106° 

46' 

40" 

B = 

46° 

53' 

14" 

C = 

26° 

20' 

6" 

Check.  yl +5  + (7  = 180°  0'  0" 


Even  if  no  mistakes  are  made  in  the  work,  the  sum  of  the  three  angles  found 
as  above  may  difier  very  slightly  from  180°  in  consequence  of  the  fact  that 
computation  with  logarithms  is  at  best  only  a method  of  close  approximation. 
When  a difference  of  this  kind  exists,  it  should  be  divided  among  the  angles 
according  to  the  probable  amount  of  error  for  each  angle. 


THE  OBLIQUE  TRIAHOLE 


m 


Exercise  55.  Finding  the  Angles 


Find  the  three  angles  of  a triangle,  given  the  three  sides  as  follows: 

1.  51,  65,  20.  6.  43,  50,  57.  11.  6,  8,  10. 

2.  78,  101,  29.  7.  37,  58,  79.  12.  6,  6,  10. 

3.  Ill,  145,  40.  8.  73,  82,  91.  13.  6,  6,  6. 

4.  21,  26,  31.  9.  Vs,  V6,  V7.  14.  6,  9,  12. 

5.  19,  34,  49.  10.  21,  28,  35.  15.  3,  4,  5. 

16.  Given  a =14.5,  b = 55.4,  and  c = 66.9,  find  A,  B,  and  C. 

17.  Given  a = 2,h  = V6,  and  c =V3  — 1,  find  A,  B,  and  C. 

18.  Given  a = 2,  b = VO,  and  c = VS  + 1,  find  A,  B,  and  C. 

19.  The  sides  of  a triangle  are  78.9,  65.4,  and  97.3  respectively. 

Find  the  largest  angle. 

20.  The  sides  of  a triangle  are  487.25,  512.33,  and  544.37  respec- 
tively. Find  the  smallest  angle. 

T.-  n ^ 1 1 Vs+i  Vs— 1 

21.  Find  the  angles  of  a triangle  whose  sides  are j =-j 

VI 

and  respectively. 


22.  Of  three  towns.  A,  B,  and  C,  A is  found  to  be  200  mi.  from  B 
and  184  mi.  from  C,  and  B is  found  to  be  150  mi.  due  north  from  C. 
How  many  miles  is  A north  of  C ? 

23.  Under  what  visual  angle  is  an  object  7 ft.  long  seen  by  an 
observer  whose  eye  is  5 ft.  from  one  end  of  the  object  and  8 ft.  from 
the  other  end  ? 

24.  The  sides  of  a triangle  are  14.6  in.,  16.7  in.,  and  18.8  in. 
respectively.  Find  the  length,  of  the  perpendicular  from  the  vertex 
of  the  largest  angle  upon  the  opposite  side. 

25.  The  distances  between  three  cities.  A,  B,  and  C,  are  measui’ed 
and  found  to  be  as  follows:  .45=165  mi.,  AC  = 72  mi.,  and 
BC  = 185  mi.  B is  due  east  from  A.  In  what  direction  is  C from  A ? 
What  two  answers  are  admissible  ? 


26.  In  a quadrilateral  ABCD,  AB  = 2 in.,  BC  = 3 in.,  CD  = 3 in., 
DA  = 4 in.,  and  AC  = ^ in.  Find  the  angles  of  the  quadrilateral. 

27.  In  a parallelogram  ABCD,  AB  = 2m.,  AC  = 3 in.,  and  AD 
= 2.5  in.  Find  Z CBA. 

28.  In  a rectangle  ABCD,  AB  = 3.3  in.,  and  A C = 5^  in.  Find  the 
angles  that  each  diagonal  makes  with  the  sides. 


128 


PLANE  TKIGONOMETRY 


118.  Area  of  a Triangle.  The  area  of  a triangle  may  be  found  if 
the  following  parts  are  known  : 

1.  Two  sides  and  the  included  angle ; 

2.  Two  angles  and  any  side  ; 

3.  The  three  sides. 

These  cases  will  now  be  considered. 

Case  1.  Given  two  sides  and  the  included  angle. 

Lettering  the  triangle  as  here  shown,  and  designating  CZ>  by  A 
and  the  area  by  5,  we  have 

6’  = ^ ch. 

But  h — a sin  B. 

Therefore  S = ^ac  sin  B. 

Also  S = ^ab  sin  C,  and  S = ^hc  sin  A. 

Exercise  56.  Area  of  a Triangle 

Find  the  areas  of  the  triangles  in  which  it  is  given  that ; 


1. 

1-7 

II 

c = 32, 

B = 40°. 

2. 

a = 35, 

c = 43. 

B = 37°. 

3. 

a = 4.8, 

c — 0.3, 

B = 39°  27'. 

4. 

a = 9.8, 

c = 7.6, 

B = 48.5°. 

6. 

a = 17.3, 

b = 19.4, 

C = 56.25°. 

6. 

a = 48.35, 

b = 64.32, 

C = 62°  37'. 

7. 

h = 127.8, 

c = 168.5, 

A = 72°  43'. 

8. 

b = 423.9, 

c = 417.8, 

A = 68°  27'. 

9. 

b = 32.78, 

c = 29.62, 

A = 57°  32'  20". 

10. 

b = 1487, 

c = 1634, 

A = 61°  30'  30". 

11.  Prove  that  the  area  of  a parallelogram  is  equal  to  the  product 
of  the  base,  the  diagonal,  and  the  sine  of  the  angle  included  by  them. 

12.  Eind  the  area  of  the  quadrilateral  ABCD,  given  AB  = 3 in., 
JC  = 4.2  in.,  .4L»=3.8  in.,  Z£dD  = 88°  10',  ZA.4C  = 36° 20'. 

13.  In  a quadrilateral  ABCD,  BC  = 5.1  in.,  ^1C  = 4.8  in.,  CD  = 
3.7  in.,  Z. A CB  = 123°  4:2',  and  ZDCA  =117°  2&.  Draw  the  figure 
approximately  and  find  the  area. 

14.  In  the  pentagon  ABODE,  AB  = 3.1  in.,  .4C  = 4.2  in.,  AD  = 
3.7  in.,  2.9  in.,  Z.4  =132°  18',  Z5.4C  = 38°  16',  and  ZDAE  = 
53°  9'.  Find  the  area  of  the  pentagon. 


THE  OBLIQUE  TRIANGLE 


129 


Case  2.  Given  two  angles  and  any  side. 

If  two  angles  are  known  the  third  can  be  found,  so  we  may 
consider  that  all  three  angles  are  given. 


G 


it  follows  that 
And  since 
we  have 


a sin  C 
c = —. — — • 
sind 

S = ^ac  sinR  (Case  1), 


= 


asinC  . „ sin  2?  sin  (7 

— — smA  = — ^ — 

Sind  2 Sind 


Since  all  three  angles  are  known  we  may  use  this  formula;  or, 
since  sin (R  -f-  C)  = sin (180°  — d)  = sind,  we  may  write  it  as  follows : 


sin  B sin  C 
2 sin  (B  + C) 


Exercise  57.  Area  of  a Triangle 

Find  the  areas  of  the  triangles  in  which  it  is  given  that : 


1. 

a=n, 

B = 

48°, 

C = 

52°. 

2. 

a - 182, 

B = 

63.5°, 

C = 

78.4°. 

3. 

a = 298, 

B = 

78.8°, 

C = 

95.5°. 

4. 

a = 19.8, 

5 = 

39°  20', 

88°  40'. 

5. 

a = 2487, 

87°  28', 

69°  32'. 

6. 

b - 483.7, 

d = 

84°  32', 

C = 

78°  49'. 

7. 

h = 527.4, 

d = 

73°  42', 

63°  37'. 

8 

c = 296.3, 

A = 

58°  35', 

B - - 

42°  36'. 

9. 

c = 17.48, 

d = 

36°  27'  30", 

B = 

73°  50'. 

10. 

c = 96.37, 

d = 

42°  23'  35", 

B = 

69°  52'  50". 

11.  In  a parallelogram  ABCD  the  diagonal  AC  makes  with  the 
sides  the  angles  27°  10'  and  32°  43'  respectively.  AB  is  2.8  in.  long. 
What  is  the  area  of  the  parallelogram  ? . , 


130 


PLANE  TRIGONOMETRY 


Case  3.  Given  the  three  sides. 

Since,  by  § 101,  sinR  = 2 sin  cos 


and,  by  § 115, 


and 


sin  ^ 


(s  — g)  (s  — c) 


ac 


cos  \B  = 


N 


by  substituting  these  values  for  sin-|-R  and  cos  in  the  above 
equation,  we  have 

2 , 

sinR  = — Vs(s  - a){s~b){s  - c). 


By  putting  this  value  for  sin  R in  the  formula  of  Case  1,  we  have 
the  following  important  formula  for  the  area  of  a triangle : 

S = ^ s(s  — d)(s  — ft)  (s  — c). 

This  is  known  as  Heron’s  Formula  for  the  area  of  a triangle,  having  been 
given  in  the  works  of  this  Greek  writer.  It  is  often  given  in  geometry,  but  the 
proof  by  trigonometry  is  much  simpler. 


A special  case  of  finding  the  area  of  a triangle  when  the  three 
sides  are  given  is  that  in  which  the  radius  of  the  circumscribed 
circle  or  the  radius  of  the  inscribed  circle  is  also  given. 

If  R denotes  the  radius  of  the  circumscribed  circle,  we  have. 


If  r denotes  the  radius  of  the  inscribed  circle,  we  may  divide  the 
triangle  into  three  triangles  by  lines  from  the  center  of  this  circle  to 
the  vertices  ; then  the  altitude  of  each  of  the  three  triangles  is  equal 

to  r.  Therefore  , 

S = ir(a  + 6 + c)  = rs. 


By  putting  in  this  formula  the  value  of  S from  Heron’s  Formula, 
we  have 


(s  — a)  (s  — b)  (s  — c) 


From  this  formula,  7-,  as  given  in  § 116,  is  seen  to  be  equal  to  the 
radius  of  the  inscribed  circle. 


THE  OBLIQUE  TRIANGLE 


131 


Exercise  58.  Area  of  a Triangle 


Find  the  areas  of  the  triangles  in  which  it  is  given  that : 


1. 

a = 3, 

11 

c = 5. 

4. 

a - 1.8, 

CO 

II 

II 

2. 

a = 15, 

b = 20, 

c = 25. 

5. 

a = 5.3, 

b = 4.8, 

c = 4.6. 

3. 

a = 10, 

b = 10, 

O 
1— ( 

II 

6. 

a = 7.1, 

b = 5.3, 

c = 6.4. 

7.  There  is  a triangular  piece  of  land  with  sides  48.5  rd.,  52.3  rd., 
and  61.4  rd.  Find  the  area  in  square  rods  ; in  acres. 

Find  the  areas  of  the  triangles  in  which  it  is  given  that : 


8. 

a = 2.4, 

b - 3.2, 

c = 4, 

R = 2. 

9. 

a = 2.7, 

CO 

II 

c = 4.5, 

R = 2.25. 

10. 

a — 3.9, 

b = 5.2, 

c = 6.5, 

R = 3.25. 

11. 

a =12, 

II 

c=12, 

R = 6.928. 

12.  Given  a = 60,  A = 40°  35' 12",  area  =12,  find  the  radius  of 
the  inscribed  circle. 

Find  the  areas  of  the  triangles  in  which  it  is  given  that : 


13. 

a - 40, 

b = 13, 

c = 37. 

14. 

a = 408, 

II 

c = 401. 

15. 

a = 624, 

b = 205, 

c = 445. 

16. 

II 

c = 5, 

O * 

O 

II 

17. 

a = l, 

e = 3, 

A = 60°. 

18. 

b - 21.66, 

c = 36.94, 

A = 66°  4' 19". 

19. 

a = 215.9, 

c = 307.7, 

A = 25°  9'  31". 

20. 

b = 149, 

A = 70°  42'  30", 

B = 39°  18'  28". 

21. 

a = 4474.5, 

b = 2164.5, 

C = 116°  30'  20". 

22. 

a = 510, 

c = 173, 

B = 162°  30'  28". 

23.  If  a is  the  side  of  an  equilateral  triangle,  show  that  the  area 
is  1 a^  Vs. 

24.  Two  sides  of  a triangle  are  12.38  ch.  and  6.78  eh.,  and  the 
included  angle  is  46°  24'.  Find  the  area. 

25.  Two  sides  of  a triangle  are  18.37  ch.  and  13.44  ch.,  and  they 
form  a right  angle.  Find  the  area. 

26.  Two  angles  of  a triangle  are  76°  54'  and  57°  33' 12",  and  the 
included  side  is  9 ch.  Find  the  area. 

27.  The  three  sides  of  a triangle  are  49  eh.,  50.25  ch.,  and  25.69  ch. 
Find  the  area. 


132 


PLANE  TEIGONOMETKY 


28.  The  three  sides  of  a triangle  are  10.64  ch.,  12.28  ch.,  and 
9 ch.  Find  the  area. 

29.  The  sides  of  a triangular  field,  of  which  the  area  is  14  A., 
are  proportional  to  3,  5,  7.  Find  the  sides. 

30.  Two  sides  of  a triangle  are  19.74  ch.  and  17.34  ch.  The  first 
bears  N.  82°  30'  W. ; the  second  S.  24°  15'  E.  Find  the  area. 

31.  The  base  of  an  isosceles  triangle  is  20,  and  its  area  is 
100  Vs ; find  its  angles. 

32.  Two  sides  and  the  included  angle  of  a triangle  are  2416  ft., 
1712  ft.,  and  30°;  and  two  sides  and  the  included  angle  of  another 
triangle  are  1948  ft.,  2848  ft.,  and  150°.  Find  the  sum  of  their  areas. 

33.  Two  adjacent  sides  of  a rectangle  are  52.25  ch.  and  38.24  ch. 
Find  the  area. 

34.  Two  adjacent  sides  of  a parallelogram  are  59.8  ch.  and  37.05  ch., 
and  the  included  angle  is  72°  10'.  Find  the  area. 

35.  Two  adjacent  sides  of  a parallelogram  are  15.36  ch.  and 
11.46  ch.,  and  the  included  angle  is  47°  30'.  Find  the  area. 

36.  Show  that  the  area  of  a quadrilateral  is  equal  to  one  half  the 
product  of  its  diagonals  into  the  sine  of  the  included  angle. 

37.  The  diagonals  of  a quadrilateral  are  34  ft.  and  56  ft.,  inter- 
secting at  an  angle  of  67°.  Find  the  area. 

38.  The  diagonals  of  a quadrilateral  are  75  ft.  and  49  ft.,  inter- 
secting at  an  angle  of  42°.  Find  the  area. 

39.  In  the  quadrilateral  A A CD  we  have  A A,  17.22  ch.;  AD,  7.45  ch.; 
CD,  14.10  ch.;  DC,  5.25  ch. ; and  the  diagonal  AC,  15.04  ch.  Find 
the  area. 

40.  Show  that  the  area  of  a regular  polygon  of  n sides,  of  which 

. . na^  180° 

one  side  is  a,  is  — — cot 

4 n 

41.  One  side  of  a regular  pentagon  is  25.  Find  the  area. 

42.  One  side  of  a regular  hexagon  is  32.  Find  the  area. 

43.  One  side  of  a regular  decagon  is  46.  Find  the  area. 

44.  If  r is  the  radius  of  a circle,  show  that  the  area  of  the  regular 

180° 

circumscribed  polygon  of  n sides  is  ni^  tan  ^ > and  the  area  of  the 
regular  inscribed  polygon  is  ^ sin  ~~~  • 

45.  Obtain  a formula  for  the  area  of  a parallelogram  in  terms  of 
two  adjacent  sides  and  the  included  angle. 


CHAPTER  VIII 


MISCELLANEOUS  APPLICATIONS 

119.  Applications  of  the  Right  Triangle.  Although  the  formulas 
for  oblique  triangles  apply  with  equal  force  to  right  triangles,  yet 
the  formulas  developed  for  the  latter  in  Chapter  IV  are  somewhat 
simpler  and  should  be  used  when  possible.  It  will  be  remembered  that 
these  formulas  depend  merely  on  the  definitions  of  the  functions. 


Exercise  59.  Right  Triangles 


1.  If  the  sun’s  altitude  is  30°,  find  the  length  of  the  longest 
shadow  which  can  be  cast  on  a horizontal  plane  by  a 
stick  10  ft.  in  length. 

2.  A flagstaff  90  ft.  high,  on  a horizontal  plane, 
casts  a shadow  of  117  ft.  Pind  the  altitude  of  the  sun. 


3.  If  the  sun’s  altitude  is  60°,  what  angle  must  a stick  make  with 
the  horizon  in  order  that  its  shadow  in  a horizontal  plane  may  be 
the  longest  possible  ? 

4.  A tower  93.97  ft.  high  is  situated  on  the  bank  of  a river.  The 
angle  of  depression  of  an  object  on  the  opposite 
bank  is  25°  12'.  Find  the  breadth  of  the  river. 

6.  The  angle  of  elevation  of  the  top  of  a tower 
is  48°  19',  and  the  distance  of  the  base  from  the  point  of  obser- 
vation is  95  ft.  Find  the  height  of  the  tower  and  the  distance  of  its 
top  from  the  point  of  observation. 


6.  From  a tower  58  ft.  high  the  angles  of  depression  of  two 

objects  situated  in  the  same  horizontal  line  with  

the  base  of  the  tower,  and  on  the  same  side,  are 

30°  13'  and  45°  46'.  Find  the  distance  between ^ ^ 

these  two  objects. 

7.  From  one  edge  of  a ditch  36  ft.  wide  the  angle  of  elevation 
of  the  top  of  a wall  on  the  opposite  edge  is  62°  39'.  Find  the 
length  of  a ladder  that  wiU  just  reach  from  the  point  of  observation 
to  the  top  of  the  wall. 


133 


134 


PLANE  TRIGONOMETEY 


8.  The  top  of  a flagstaff  has  been  partly  broken  off  and  touches 
the  ground  at  a distance  of  15  ft.  from  the  foot  of  the  staff.  If  the 
length  of  the  broken  part  is  39  ft.,  find  the  length  of  the  whole  staff. 

9.  From  a balloon  which  is  directly  above  one  town  the  angle 
of  depression  of  another  town  is  observed  to  be  10°  14'.  The  towns 
being  8 mi.  apart,  find  the  height  of  the  balloon. 

10.  A ladder  40  ft.  long  reaches  a , window  33  ft.  high,  on  one 
side  of  a street.  Being  turned  over  upon  its  foot,  the  ladder  reaches 
another  window  21  ft.  high,  on  the  opposite  side  of  the  street.  Find 
the  width  of  the  street. 

11.  From  a mountain  1000  ft.  high  the  angle  of  depression  of  a 
ship  is  27°  35' 11".  Find  the  distance  of  the  ship  from  the  summit 
of  the  mountain. 

12.  From  the  top  of  a mountain  3 mi.  high  the  angle  of  depres- 
sion of  the  most  distant  object  which  is  visible  on  the  earth’s  sur- 
face is  found  to  be  2°  13'  50".  Find  the  diameter  of  the  earth. 

13.  A lighthouse  54  ft.  high  is  situated  on  a rock.  The  angle  of 
elevation  of  the  top  of  the  lighthouse,  as  observed  from  a ship,  is 
4°  52',  and  the  angle  of  elevation  of  the  top  of  the  rock  is  4°  2'. 
Find  the  height  of  the  rock  and  its  distance  from  the  ship. 

14.  The  latitude  of  Cambridge,  Massachusetts,  is  42°  22' 49".  What 
is  the  length  of  the  radius  of  that  parallel  of  latitude  ? 

15.  At  what  latitude  is  the  circumference  of  the  parallel  of  lati- 
tude equal  to  half  the  equator  ? 

16.  In  a circle  with  a radius  of  6.7  is  inscribed  a regular  polygon 
of  thirteen  sides.  Find  the  length  of  one  of  its  sides. 

17.  A regular  heptagon,  one  side  of  which  is  5.73,  is  inscribed  in 
a circle.  Find  the  radius  of  the  circle. 

18.  When  the  moon  is  setting  at  any  place,  the  angle  at  the  mooir 
subtended  by  the  earth’s  radius  passing  through  that  place  is  57'  3". 
If  the  earth’s  radius  is  3956.2  mi.,  what  is  the  moon’s  distance  from 
the  earth’s  center  ? 

19.  A man  in  a balloon  observes  the  angle  of  depression  of  an 
object  on  the  ground,  bearing  south,  to  be  35°  30';  the  balloon  drifts 
2\  mi.  east  at  the  same  height,  when  the  angle  of  depression  of  the 
same  object  is  23°  14'.  Find  the  height  of  the  balloon. 

20.  The  angle  at  the  earth’s  center  subtended  by  the  sun’s  radius 
is  16'  2",  and  the  sun’s  distance  is  92,400,000  mi.  Find  the  sun’s 
diameter  in  miles. 


MISCELLANEOUS  APPLICATIONS 


135 


21.  A man  standing  south  of  a tower  and  on  the  same  horizontal 
plane  observes  its  angle  of  elevation  to  be  54°  16';  he  goes  east 
100  yd.  and  then  finds  its  angle  of  elevation  is  50°  8'.  Find  the 
height  of  the  tower. 

22.  A regular  pyramid,  with  a square  base,  has  a lateral  edge  150  ft. 
long,  and  the  side  of  the  base  is  200  ft.  Find  the  inclination  of  the 
face  of  the  pyramid  to  the  base. 

23.  The  height  of  a house  subtends  a right  angle  at  a window  on 
the  other  side  of  the  street,  and  the  angle  of  elevation  of  the  top  of 
the  house  from  the  same  point  is  60°.  The  street  is  30  ft.  wide. 
How  high  is  the  house  ? 

24.  The  perpendicular  from  the  vertex  of  the  right  angle  of  a 
right  triangle  divides  the  hypotenuse  into  two  segments  364.3  ft. 
and  492.8  ft.  in  length  respectively.  Find  the  acute  angles  of  the 
triangle. 

26.  The  bisector  of  the  right  angle  of  a right  triangle  divides  the 
hypotenuse  into  two  segments  431.9  ft.  and  523.8  ft.  in  length 
respectively.  Find  the  acute  angles  of  the  triangle. 

26.  Find  the  number  of  degrees,  minutes,  and  seconds  in  an  arc 
of  a circle,  knowing  that  the  chord  which  subtends  it  is  238.25  ft., 
and  that  the  radius  is  196.27  ft. 

27.  Calculate  to  the  nearest  hundredth  of  an  inch  the  chord  which 
subtends  an  arc  of  37°  43'  in  a circle  having  a radius  of  542.35  in. 

28.  Calculate  to  the  nearest  hundredth  of  an  inch  the  chord  which 
subtends  an  arc  of  14°  in  a circle  having  a radius  of  47 5.23  in. 

29.  In  an  isosceles  triangle  ABC  the  base  AB  is  1235  in.,  and 
Z.4  = ZS  = 64°  22'.  Find  the  radius  of  the  inscribed  circle. 

30.  Find  the  number  of  degrees,  minutes,  and  seconds  in  an  arc 
of  a circle,  knowing  that  the  chord  which  subtends  it  is  two  thirds 
of  the  diameter. 

31.  Find  the  number  of  degrees,  minutes,  and  seconds  in  an  arc 
of  a circle,  knowing  that  the  chord  which  subtends  it  is  three  fourths 
of  the  diameter. 

32.  The  radius  of  a circle  being  2548.36  in.,  and  the  length  of  a 
chord  BC  being  3609.02  in.,  find  the  angle  BAC  made  by  two 
tangents  drawn  at  B and  C respectively. 

33.  Find  the  ratio  of  a chord  to  the  diameter,  knowing  that  the 
chord  subtends  an  arc  27°  48'.  If  the  diameter  is  8 in.,  how  long  is 
the  chord  ? If  the  chord  is  8 in.,  how  long  is  the  diameter  ? 


136 


PLANE  TEIGONOMETRY 


34.  Find  the  length  of  the  diameter  of  a regular  pentagon  of 
which  the  side  is  1 in.,  and  the  length  of  the  side  of  a regular 
pentagon  of  which  the  diameter  is  1 in. 


36.  Two  circles  of  radii  a and  h are  externally  tangent.  The  com- 
mon tangents  AP,  BP,  and  the  line  of  centers  CC’P  are  drawn  as 
shown  in  the  figure.  Find  sin  APC. 

36.  In  Ex.  35  find  /.APC,  know- 
ing that  a — 2>b. 

37.  In  ZA=  68°26'27", 

/B  = 75°  8'  23",  and  the  altitude  h, 
from  C,  is  148.17  in.  Eequired  the 
lengths  of  the  three  sides. 


38.  Two  axes,  OX  and  OY,  form  a right  angle  at  0,  the  center  of 
a circle  of  radius  1091  ft.  Through  P,  a point  on  OX  1997  ft.  from 
0,  a tangent  is  drawn,  meeting  OF  at  C.  Ee- 
quired OC  and  the  angle  CPO. 

39.  Find  the  sine  of  the  angle  formed  by 
the  intersection  of  the  diagonals  of  a cube. 

40.  The  angle  of  elevation  of  the  top  of 
a tower  observed  at  a place  A,  south  of  it,  is 
30° ; and  at  a place  B,  west  of  A , and  at  a distance  of  a from  it, 
the  angle  of  elevation  is  18°.  Show  that  the  height  of  the  tower 

is  ~~r=^ , the  tangent  of  18°  being ~ j: 

'^2  + 2V6  ViFTTv! 


41.  Standing  directly  in  front  of  one  corner  of  a flat-roofed  house, 
which  is  150  ft.  in  length,  I observe  that  the  horizontal  angle  which 
the  length  subtends  has  for  its  cosine  V^,  and  that  the  vertical  angle 

3 

subtended  by  its  height  has  for  its  sine  n—  • What  is  the  height 
of  the  house  ? 


42.  At  a distance  a from  the  foot  of  a tower,  the  angle  of  eleva- 
tion A of  the  top  of  the  tower  is  the  complement  of  the  angle  of 
elevation  of  a flagstaff  on  top  of  it.  Show  that  the  length  of  the 
staff  is  2 a cot  2 A. 


43.  A rectangular  solid  is  4 in.  long,  3 in.  wide,  and  2 in.  high. 
Calculate  the  tangent  of  the  angle  formed  by  the  intersection  of 
any  two  of  the  diagonals. 

44.  Calculate  the  tangent  as  in  Ex.  43,  the  solid  being  I units  long, 
w units  wide,  and  A units  higln 


MISCELLANEOUS  APPLICATIONS 


137 


120.  Applications  of  the  Oblique  Triangle.  As  stated  in  § 119,  when 
conditions  permit  of  using  a right  triangle  in  making  a trigono- 
metric observation  it  is  better  to  do  so.  Often,  however,  it  is  impos- 
sible or  inconvenient  to  use  the  right  triangle,  as  in  the  case  of  an 
observation  on  an  inclined  plane,  and  in  such  cases  resort  to  the 
oblique  triangle  is  necessary. 

Exercise  60.  Oblique  Triangles 

1.  Show  how  to  determine  the  height  of  an  inaccessible  object 
situated  on  a horizontal  plane  by  observing  its  angles  of  elevation 
at  two  points  in  the  same  line  with  its  base  and  measuring  the 
distance  between  these  two  points. 

2.  Show  how  to  determine  the  height  of  an  inaccessible  object 
standing  on  an  inclined  plane. 

3.  Show  how  to  determine  the  distance  between  two  inaccessible 
objects  by  observing  angles  at  the  ends  of  a line  of  known  length. 

4.  The  angle  of  elevation  of  the  top  of  an  inaccessible  tower  stand- 
ing on  a horizontal  plain  is  63°  26' ; at  a point  500  ft.  farther  from 
the  base  of  the  tower  the  angle  of  elevation  of  the  top  is  32°  14'. 
Find  the  height  of  the  tower. 

5.  A tower  stands  on  the  bank  of  a river.  From  the  opposite  bank 
the  angle  of  elevation  of  the  top  of  the  tower  is  60°  13',  and  from  a 
point  40  ft.  further  off  the  angle  of  elevation  is  50°  19'.  Find  the 
width  of  the  river. 

6.  At  the  distance  of  40  ft.  from  the  foot  of  a vertical  tower  on 
an  inclined  plane,  the  tower  subtends  an  angle  of  41°  19';  at  a point 
60  ft.  farther  away  the  angle  subtended  by  the  tower  is  23°  45'. 
Find  the  height  of  the  tower. 

7.  A building  makes  an  angle  of  113°  12'  with  the  inclined  plane 

on  which  it  stands ; at  a distance  of  89  ft.  from  its  base,  measured 
down  the  plane,  the  angle  subtended  by  the  building  is  23°  27'.  Find 
the  height  of  the  building.  

8.  A person  goes  70  yd.  up  a slope  of  1 in  3^  from  the  bank  of  a 
river  and  observes  the  angle  of  depression  of  an  object  on  the  oppo- 
site bank  to  be  2i°.  Find  the  width  of  the  river. 

9.  A tree  stands  on  a declivity  inclined  15°  to  the  horizon.  A man 
ascends  the  declivity  80  ft.  from  the  foot  of  the  tree  and  finds  the 
angle  then  subtended  by  the  tree  to  be  30°.  Find  the  height  of 
the  tree. 


138 


PLANE  TRIGONOMETRY 


10.  The  angle  subtended  by  a tree  on  an  inclined  plane  is,  at  a 
certain  point,  42°  17',  and  325  ft.  further  down  it  is  21°  47'.  The 
inclination  of  the  plane  is  8°  53'.  Find  the  height  of  the  tree. 

11.  From  a point  B at  the  foot  of  a mountain,  the  angle  of  elevation 
of  the  top  A is  60°.  After  ascending  the  mountain  one  mile,  at  an 
inclination  of  30°  to  the  horizon,  and  reaching  a point  C,  an  observer 
finds  that  the  angle  A CB  is  135°.  Find  the  number  of  feet  in  the 
height  of  the  mountain. 

12.  The  length  of  a lake  subtends,  at  a certain  point,  an  angle  of 
46°  24',  and  the  distances  from  this  point  to  the  two  ends  of  the 
lake  are  346  ft.  and  290  ft.  Find  the  length  of  the  lake. 

13.  Along  the  bank  of  a river  is  drawn  a base  line  of  500  ft.  The 
angular  distance  of  one  end  of  this  line  from  an  object  on  the  oppo- 
site side  of  the  river,  as  observed  from  the  other  end  of  the  line, 
is  53°;  that  of  the  second  extremity  from  the  same  object,  observed 
at  the  first,  is  79°  12'.  Find  the  width  of  the  river. 

14.  Two  observers,  stationed  on  opposite  sides  of  a cloud,  observe 
its  angles  of  elevation  to  be  44°  56'  and  36°  4'.  Their  distance  from 
each  other  is  7 00  ft.  What  is  the  height  of  the  cloud  ? 

15.  From  the  top  of  a house  42  ft.  high  the  angle  of  elevation  of 
the  top  of  a pole  is  14°  13';  at  the  bottom  of  the  house  it  is  23°  19'. 
Find  the  height  of  the  pole. 

16.  From  a window  on  a level  with  the  bottom  of  a steeple  the 
angle  of  elevation  of  the  top  of  the  steeple  is  40°,  and  from  a second 
window  18  ft.  higher  the  angle  of  elevation  is  37°  30'.  Find  the 
height  of  the  steeple. 

17.  The  sides  of  a triangle  are  17,  21,  28.  Prove  that  the  length 
of  a line  bisecting  the  longest  side  and  drawn  from  the  opposite 
angle  is  13. 

18.  The  sum  of  the  sides  of  a triangle  is  100.  The  angle  at  is 
double  that  at  B,  and  the  angle  at  B is  double  that  at  C.  Determine 
the  sides. 

19.  A ship  sailing  north  sees  two  lighthouses  8 mi.  apart  in  a line 
(iue  west ; after  an  hour’s  sailing,  one  lighthouse  bears  S.W.,  and 
the  other  S.  22°  30'  W.  (22°  30'  west  of  south).  Find  the  ship’s  rate. 

20.  A ship,  10  mi.  S.W.  of  a harbor,  sees  another  ship  sail  from 
the  harbor  in  a direction  S.  80°  E.,  at  a rate  of  9 mi.  an  hour.  In  what 
direction  and  at  what  rate  must  the  first  ship  sail  in  order  to  catch 
up  with  the  second  ship  in  1^  hr.  ? 


MISCELLAITEOUS  APPLICATIONS 


139 


21.  Two  ships  are  a mile  apart.  The  angular  distance  of  the  first 
ship  from  a lighthouse  on  shore,  as  observed  from  the  second  ship, 
is  35°  14'  10" ; the  angular  distance  of  the  second  ship  from  the  light- 
house, observed  from  the  first  ship,  is  42°  11'  53".  Find  the  distance 
in  feet  from  each  ship  to  the  lighthouse. 

22.  A lighthouse  bears  N.  11°  15'  E.,  as  seen  from  a ship.  The 
ship  sails  northwest  30  mi.,  and  then  the  lighthouse  bears  east.  How 
far  is  the  lighthouse  from  the  second  point  of  observation  ? 

23.  Two  rocks  are  seen  in  the  same  straight  line  with  a ship, 
bearing  N.  15°  E.  After  the  ship  has  sailed  N.W.  5 mi.,  the  first  rock 
bears  E.,  and  the  second  N.E.  Find  the  distance  between  the  rocks. 

24.  On  the  side  OX  of  a given  angle  XOY  a point  A is  taken  such 
that  OA  = d.  Deduce  a formula  for  the  length  AS  of  a line  from  A 
to  0 F that  makes  a given  angle  a with  OX.  From 
this  formula,  a;  is  a minimum  when  what  sine  is 
the  maximum?  Under  those  circumstances  what 
is  the  sum  of  0 and  a ? Then  what  is  the  size  of 
ZS?  State  the  conclusion  as  to  the  size  of  Zu 
in  order  that  x shall  be  the  minimum. 

25.  Three  points.  A,  B,  and  C,  form  the  vertices  of  an  equilateral 
triangle,  AB  being  500  ft.  Each  of  the  two  sides  AB  and  AC  is  seen 
from  a point  P under  an  angle  of  120° ; that  is,  Z A PB  = 120°=Z  CPA. 
Find  the  length  of  AP. 

[/26.  A lighthouse  facing  south  sends  out  its  rays  extending  in  a 
quadrant  from  S.E.  to  S.W.  A steamer  sailing  due  east  first  sees 
the  light  when  6 mi.  away  from  the  lighthouse  and  continues  to  see 
it  for  45  min.  At  what  rate  is  the  ship  sailing  ? 

27.  If  two  forces,  represented  in  intensity  by  the  lengths  a and  b, 
pull  from  P in  the  directions  PC  and  PA,  respectively,  and  if  Z.APC 
is  known,  the  resultant  force  is  represented  in 
intensity  and  direction  by  f,  the  diagonal  of 
the  parallelogram  ABCP.  Show  how  to  find/ 
and  A APB,  given  a,  b,  and  AAPC. 

'y28.  Two  forces,  one  of  410  lb.  and  the  other 
of  320  lb.,  make  an  angle  of  51°  37'.  Find  the  intensity  and  the 
direction  of  their  resultant. 

29.  An  unknown  force  combined  with  one  of  128  lb.  produces 
a resultant  of  200  lb.,  and  this  resultant  makes  an  angle  of  18° 
24'  with  the  known  force.  Find  the  intensity  and  direction  of  the 
unknown  force. 


140 


PLANE  TRIGONOMETEY 


30.  Wishing  to  determine  the  distance  between  a church  A and  a 
tower  B,  on  the  opposite  side  of  a river,  a man  measured  a line  CD 
along  the  river  ((7  being  nearly  opposite  A),  and  observed  the  angles 
ACB,  58°  20';  ACD,  95°  20';  ADB,  53°  30';  BDC,  98°  45'.  CD  is 
600  ft.  What  is  the  distance  required  ? 

31.  Wishing  to  find  the  height  of  a summit  A,  a man  measured  a 
horizontal  base  line  CD,  440  yd.  At  C the  angle  of  elevation  of  A 
is  37°  18',  and  the  horizontal  angle  between  D and  the  summit  of 
the  mountain  is  76°  18';  at  D the  horizontal  angle  between  C and 
the  summit  is  67°  14'.  Eind  the  height. 

32.  A balloon  is  observed  from  two  stations  3000  ft.  apart.  At  the 
first  station  the  horizontal  angle  of  the  balloon  and  the  other  station 
is  75°  25',  and  the  angle  of  elevation  of  the  balloon  is  18°.  The  hori- 
zontal angle  of  the  first  station  and  the  balloon,  measured  at  the 
second  station,  is  64°  30'.  Find  the  height  of  the  balloon. 

33.  At  two  stations  the  height  of  a kite  subtends  the  same  angle  A. 
The  angle  which  the  line  joining  one  station  and  the  kite  subtends 
at  the  other  station  is  B ; and  the  distance  between  the  two  stations 
is  a.  Show  that  the  height  of  the  kite  is  ^ a sin  A sec  B. 

34.  Two  towers  on  a horizontal  plain  are  120  ft.  apart.  A person 
standing  successively  at  their  bases  observes  that  the  angle  of  eleva- 
tion of  one  is  double  that  of  the  other ; but  when  he  is  halfway  be- 
tween the  towers,  the  angles  of  elevation  are  complementary.  Prove 
that  the  heights  of  the  towers  are  90  ft.  and  40  ft. 

35.  To  find  the  distance  of  an  inaccessible  point  C from  either 
of  two  points  A and  B,  having  no  instruments  to  measure  angles. 
Prolong  CA  to  a,  and  CB  to  b,  and  draw  AB,  Ah,  and  Ba.  Measure 
AB,  500  ft.;  aA,  100  ft.;  aB,  560  ft.;  bB,  100  ft.;  and  A 5,  550  ft. 
Compute  the  distances  A C and  B C. 

36.  To  compute  the  horizontal  distance  between  two  inaccessible 
points  A and  B when  no  point  can  be  found  whence  both  can  be  seen. 
Take  two  points  C and  D,  distant  200  yd.,  so  that  A can  be  seen 
from  C,  and  B from  D.  From  C measure  CF,  200  yd.  to  F,  whence 
A can  be  seen ; and  from  D measure  DE,  200  yd.  to  E,  whence  B 
can  be  seen.  Measure  AEC,  83°;  ACD,  53°  30';  ACF,  54°  31';  BDE, 
54°  30' ; BDC,  156°  25' ; DEB,  88°  30'.  Compute  the  distance  AB. 

37.  A column  in  the  north  temperate  zone  is  S.  67°  30'  E.  of  an 
observer,  and  at  noon  the  extremity  of  its  shadow  is  northeast  of  him. 
The  shadow  is  80  ft.  in  length,  and  the  elevation  of  the  column  at 
the  observer’s  station  is  45°.  Find  the  height  of  the  column. 


MISCELL AJJ^EOUS  APPLICATIONS 


141 


121.  Areas.  In  finding  the  areas  of  rectilinear  figures  the  effort 
is  made  to  divide  any  given  figure  into  rectangles,  parallelograms, 
triangles,  or  trapezoids,  unless  it  already  has  one  of  these  forms. 


For  example,  the  dotted  lines  show  how  the  above  figures  may  be 
divided  for  the  purpose  of  computing  the  areas.  A regular  polygon 
would  be  conveniently  divided  into  congruent  isosceles  triangles 
by  the  radii  of  the  circumscribed  circle. 


Exercise  61.  Miscellaneous  Applications 

1.  In  the  trapezoid  A jBCZ)  it  is  known  that  Z A =90°,  Z5  = 32°25', 
AB  = 324.35  ft.,  and  CD  = 208.15  ft.  Find  the  area. 

2.  Find  the  area  of  a regular  pentagon  of  which  each  side  is  4 in. 

3.  Find  the  area  of  a regrdar  hexagon  of  which  each  side  is  4 iu 

4.  The  area  of  a regular  polygon  inscribed  in  a circle  is  to  that 
of  the  circumscribed  regular  polygon  of  the  same  number  of  sides 
as  3 to  4.  Find  the  number  of  sides. 

5.  The  area  of  a regular  polygon  inscribed  in  a circle  is  the 
geometric  mean  between  the  areas  of  the  inscribed  and  circumscribed 
regular  polygons  of  half  the  number  of  sides. 

6.  Find  the  ratio  of  a square  inscribed  in  a circle  to  a square  cir- 
cumscribed about  the  same  circle.  Find  the  ratio  of  the  perimeters. 

7.  The  square  circumscribed  about  a circle  is  four  thirds  the  in- 
scribed regular  dodecagon. 

8.  In  finding  the  area  of  a field  ABODE  a surveyor  measured 
the  lengths  of  the  sides  and  the  angle  which  each  side  makes  with 
the  meridian  (north  and  south)  line  through  its 
extremities.  AD  happened  to  be  a meridian  line. 

Show  how  he  could  compute  the  area. 

9.  Two  sides  of  a triangle  are  3 and  12,  and 
the  included  angle  is  30°.  Find  the  hypotenuse  of 
the  isosceles  right  triangle  of  equal  area. 

10.  In  the  quadrilateral  A B CD  we'have  given  A B, 

BC,/-A,Z.B,  and  Z C.  Show  how  to  find  the  area  of  the  quadrilateral. 

11.  In  Ex.  10,  suppose  A5  = 175  ft.,  AC  = 198  ft.,  ZA=  95°, 
Z A = 92°  15',  and  Z C = 96°  45'.  What  is  the  area  ? 


142 


PLANE  TRIGONOMETRY 


122.  Surveyor’s  Measures.  In  measuring  city  lots  surveyors  com- 
monly use  feet  and  square  feet,  with  decimal  parts  of  these  units. 
In  measuring  larger  pieces  of  land  the  following  measures  are  used : 

16^  feet  (ft.)  = 1 rod  (rd.) 

66  feet  = 4 rods  = 1 chain  (ch.) 

100  links  (li.)  = 1 chain 

10  square  chains  (sq.  ch.)  = 160  square  rods  (sq.  rd.)= 1 acre  (A.) 

We  may  write  either  7 ch.  42  li.  or  7.42  ch.  for  7 chains  and  42  links.  The 
decimal  fraction  is  rapidly  replacing  the  old  plan,  in  which  the  word  livk  was 
used.  Similarly,  the  parts  of  an  acre  are  now  written  in  the  decimal  form 
instead  of,  as  formerly,  in  square  chains  or  square  rods. 

Areas  are  computed  as  if  the  land  were  flat,  or  projected  on  a horizontal 
plane,  no  allowance  being  made  for  inequalities  of  surface. 

123.  Area  of  a Field.  The  areas  of  fields  are  found  in  various 
ways,  depending  upon  the  shape.  In  general,  however,  the  work  is 
reduced  to  the  finding  of  the  areas  of  triangles 
or  trapezoids. 

For  example,  in  the  case  here  shown  we  may  draw  a 
north  and  south  line  E'A'  and  then  And  the  sum  of  the 
areas  of  the  trapezoids  ABB'A',  BCC'B',  CBJD'C',  and 
DEE'D' . From  this  we  may  subtract  the  sum  of  the 
trapezoids  A GG'A',  GFF'G'  a,nd  FEE'F' . The  result  will 
be  the  area  of  the  field. 

Instead  of  running  the  imaginary  line  E'A'  outside 
the  field,  it  would  be  quite  as  convenient  to  let  it  pass 
through  F,  A,  E,  or  C.  The  method  of  computing  the 
area  is  substantially  the  same  in  both  cases. 

For  details  concerning  surveying,  beyond  what  is  here  given  and  is  included 
in  Exercise  60,  the  student  is  referred  to  works  upon  the  subject. 


Exercise  62.  Area  of  a Field 

1.  Find  the  number  of  acres  in  a triangular  field  of  which  the 
sides  are  14  ch.,  16  ch.,  and  20  ch. 

2.  Find  the  number  of  acres  in  a triangular  field  having  two  sides 
16  ch.  and  30  ch.,  and  the  included  angle  64°  15'. 

3.  Find  the  number  of  acres  in  a triangular  field  having  two  angles 
68.4°  and  47.2°,  and  the  included  side  20  ch. 

4.  Required  the  area  of  the  field  described  in  § 123,  knowing  that 
AA'  = 8 ch.,  BB’  = 12  ch.,  CC  = 13  ch.,  DD'  = 12  ch.,  EE'  = 8 ch., 
FF'  = 1 ch.,  GG'  = 2 ch.,  A'G’ = 6 ch.,  G'B'  = 1.5  ch.,  B'F'  = 2.3  ch., 
F'C  = 3 ch.,  CD'  = 4 ch.,  D'E'  = 2.9  ch. 


MISCELLANEOUS  APPLICATIONS 


143 


5.  In  a quadrangialar  field  ABCD,  AB  runs  N.  27°  E.  12.5  ch., 
BC  runs  N.  30°  W.  10  cli.,  CD  runs  S.  37°  W.  15  ch.,  and  DA  runs 
S.  47°  E.  11.5  eh.  Find  the  area. 

That  AB  is  N.  27°E.  means  that  it  makes  an  angle  of 
27°  east  of  the  line  running  north  through  A. 

6.  In  a triangular  field  ABC,  AB  runs  N.  10°  E. 

30  ch.,  BC  runs  S.  30°  W.  20  ch.,  and  CA  runs  S.  22°  E. 

13  ch.  Find  the  area. 

7.  In  a field  ABCD,  AB  runs  E.  10  ch.,  BC  runs 
N.  12  ch.,  CD  runs  S.  68°  12'  W.  10.77  ch.,  and  DA 
runs  S.  8 ch.  Find  the  area. 

8.  A field  is  in  the  form  of  a right  triangle  of  which  the  sides 
are  15  ch.,  20  ch.,  and  25  ch.  From  the  vertex  of  the  right  angle  a 
line  is  run  to  the  hypotenuse,  making  an  angle  of  30°  with  the  side 
that  is  15  ch.  long.  Find  the  area  of  each  of  the  triangles  into 
which  the  field  is  divided. 


Using  a protractor,  draw  to  scale  the  fields  referred  to  in  the 
following  examples,  and  find  the  areas : 


9.  AB,  N.  72°  E.  18  ch., 
BC,  N.  10°  E.  12.5  ch.. 


10.  AB,  N.  45°  E.  10  ch., 

BC,  S.  75°  E.  11.55  ch., 

11.  AB,  N.  5°30' W.  6.08  ch., 
BC,  S.  82°  30'  W.  6.51  ch.^ 

12.  AB,  N.  6°  15'  W.  6.31  ch., 
BC,  S.  81°  50'  W.  4.06  ch.. 


CD,  N.  68°  W.  21  ch., 

DA,  S.  12°  E.  26.3  ch. 
CD,  S.  15°  W.  18.21  ch., 
DA,  N.45°  W.  19.11  ch. 

CD,  S.  3°  E.  5.33  ch., 

DA,  E.  6.72  ch. 

CD,  S.  5°  E.  5.86  ch., 

DA,  N.  88°  30'  E,  4.12  ch. 


13.  A farm  is  bounded  and  described  as  follows : Beginning  at 
the  southwest  corner  of  lot  No.  13,  thence  N.  1^°  E.  132  rods  and 
23  links  to  a stake  in  the  west  boundary  line  of  said  lot;  thence 
S.  89°  E.  32  rods  and  15-^  links  to  a stake ; thence  N.  1^°  E.  29  rods 
and  15  links  to  a stake  in  the  north  boundary  line  of  said  lot ; thence 
S.  89°  E.  61  rods  and  18  links  to  a stake ; thence  S.  32J°  W.  54  rods 
to  a stake ; thence  S.  35^°  E.  22  rods  and  4 links  to  a stake ; thence 
S.  48°  E.  33  rods  and  2 links  to  a stake ; thence  S.  7 ^°  W.  7 6 rods 
and  20  links  to  a stake  in  the  south  boundary  line  of  said  lot ; thence 
N.  89°  W.  96  rods  and  10  links  to  the  place  of  beginning.  Containing 
85.65  acres,  more  or  less.  Verify  the  area  given  and  plot  the  farm. 

TMs  is  a common  way  of  describing  a farm  in  a deed  or  a mortgage. 


144 


PLANE  TRIGONOMETRY 


124.  The  Circle.  It  is  learned  in  geometry  that 
c — 2 irr,  and  a — 

where  c = circumference,  r = radius,  a — area,  and  tt  = 3.14159+ 
= 3.1416—  = about  3i.  Eor  practical  purposes  ^ may  be  taken. 
Furthermore,  if  we  have  a sector  with  angle  n degrees, 

7t 

the  area  of  the  sector  is  evidently  of 

odU 

From  these  formulas  we  can,  by  the  help  of  the 
formulas  relating  to  triangles,  solve  numerous  prob- 
lems relating  to  the  circle. 


Exercise  63.  The  Circle 

1.  A sector  of  a circle  of  radius  8 in.  has  an  angle  of  62.5°. 
A chord  joining  the  extremities  of  the  radii  forming  the  sector  cuts 
off  a segment.  What  is  the  area  of  this  segment  ? 

2.  A sector  of  a circle  of  diameter  9.2  in.  has  an 
angle  of  29°  42'.  A chord  joining  the  extremities 
of  the  radii  forming  the  sgctor  cuts  off  a segment. 

What  is  the  area  of  the  remainder  of  the  circle  ? 

3.  In  a circle  of  radius  3.5  in.,  what  is  the  area  included  between  two 
parallel  chords  of  6 in.  and  5 in.  respectively  ? (Give  two  answers.) 

4.  A regular  hexagon  is  inscribed  in  a circle  of  radius  4 in.  What 
is  the  area  of  that  part  of  the  circle  not  covered  by  the  hexagon  ? 

5.  In  a circle  of  radius  10  in.  a regular  five-pointed 
star  is  inscribed.  What  is  the  area  of  the  star  ? What 
is  the  area  of  that  part  of  the  circle  not  covered  by 
the  star  ? 

6.  In  a circle  of  diameter  7.2  in.  a regular  five- 
pointed  star  is  inscribed.  The  points  are  joined, 
thus  forming  a regular  pentagon.  There  is  also  a regular  pentagon 
formed  in  the  center  by  the  crossing  of  the  lines  of  the  star.  The 
small  pentagon  is  what  fractional  part  of  the  large  one  ? 

7.  A circular  hole  is  cut  in  a regular  hexagonal  plate 
of  side  8 in.  The  radius  of  the  circle  is  4 in.  What  is 
the  area  of  the  rest  of  the  plate  ? 

8.  A regular  hexagon  is  formed  by  joining  the  mid-points  of  the 
sides  of  a regular  hexagon.  Find  the  ratio  of  the  smaller  hexagon 
to  the  larger. 


CHAPTER  IX 


PLANE  SAILING 

125.  Plane  Sailing.  A simple  and  interesting  application  of  plane 
trigonometry  is  found  in  that  branch,  of  navigation  in  which  the 
surface  of  the  earth  is  considered  a plane.  This  can  be  the  case 
only  when  the  distance  is  so  small  that  the  curvature  of  the  earth 
may  be  neglected. 

This  chapter  may  be  omitted  if  further  applications  of  a practical  nature  are 
not  needed. 

126.  Latitude  and  Departure.  The  difference  of  latitude  between 
two  places  is  the  arc  of  a meridian  between  the  parallels  of  latitude 
which  pass  through  those  places. 

Thus  the  latitude  of  Cape  Cod  is  42°  2'  21"  N.  and  the  latitude  of  Cape  Hat- 
teras  is  35°  15'  14"  N.  The  difference  of  latitude  is  6°  47'  7". 

The  departure  between  two  meridians  is  the  length  of  the  arc 
of  a parallel  of  latitude  cut  off  by  those  meridians,  measured  in 
geographic  miles. 

The  geographic  mile,  or  knot,  is  the  length  of  1'  of  the  equator.  Taking  the 
equator  to  he  131,385,456  ft.,  of  of  this  length  is  6082.66  ft.,  and  this 
is  generally  taken  as  the  standard  in  the  United  States.  The  British  Admiralty 
knot  is  a little  shorter,  being  6080  ft.  The  term  "mile”  in  this  chapter  refers 
to  the  geographic  mile,  and  there  are  60  mi.  in  one  degree  of  a great  circle. 

Calling  the  course  the  angle  between  the  track  of  the  ship  and  the 
meridian  line,  as  in  the  case  of  E".  20°  E.,  it  will  be  evident  by  drawing 
a figure  that  the  difference  in  latitude,  expressed  in  distance,  equals 
the  distance  sailed  multiplied  by  the  cosine  of  the  course.  That  is 
diff.  of  latitude  = distance  x cos  C. 

In  the  same  way  we  can  find  the  departure.  This  is  evidently 
given  by  the  equation 

departure  = distance  x sin  C. 

For  example,  if  a ship  has  sailed  E".  30°  E.  10  mi.,  the  difference 
in  latitude,  expressed  in  miles,  is 

10  cos  30°  = 10  X 0.8660  = 8.66, 

and  the  departure  is  10  sin  30°  = 10  x 0.5  = 5- 

145 


146 


PLANE  TRIGONOMETRY 


127.  The  Compass.  The  mariner  divides  the  circle  into  32  equal 
parts  called  points.  There  are  therefore  8 points  in  a right  angle, 
and  a point  is  11°  15'.  To  sail  two 
points  east  of  north  means,  therefore, 
to  sail  22°  30'  east  of  north,  or  north- 
northeast  (N.N.E.)  as  shown  on  the 
compass.  Northeast  (N.E.)  is  45°  east 
of  north.  One  point  east  of  north  is 
called  north  by  east  (N.  by  E.)  and  one 
point  east  of  south  is  called  south  by 
east  (S.  by  E.).  The  other  terms  used, 
and  their  significance  in  angular  measure, 
will  best  be  understood  from  the  illustration  and  the  following  table : 


North 

Points 

0-1 

0-^ 

0-i 

1 

o / // 

2 48  45 
5 37  30 
8 26  15 
11  15  0 

Points 

0-i 

0-1 

0-1 

1 

South 

N.  by  E. 

N.  by  W. 

S.  by  E. 

S.  by  W. 

N.N.E. 

N.N.W. 

1-1 

1-1 

1-1 

2 

14  3 45 
16  52  30 
19  41  15 
22  30  0 

1-i 

i-i 

2 

S.S.E. 

S.S.W. 

N.E.  by  N. 

N.W.  by  N. 

t 

3-i 

2-1 

3 

25  18  45 
28  7 30 
30  56  15 
33  45  0 

2-i 

2-i 

2-i 

3 

S.E.  by  S. 

S.W.  by  S. 

N.E. 

N.W. 

3-1 

3-3 

3-i 

4 

36  33  45 
39  22  30 
42  11  15  ' 
45  0 0 

3-i 

3-^ 

3-i 

4 

S.E. 

S.W. 

N.E.  by  E. 

N.W.  by  W. 

4-1 

4-i 

4-1 

5 

47  48  45 
50  37  30 
53  26  15 
56  15  0 

4-i 

4-i 

4-1 

5 

S.E.  by  E. 

S.W.  by  W. 

E.N.E. 

W.N.W. 

5-1 

5-1 

5-1 

6 

59  3 45 
61  52  30 
64  41  15 
67  30  0 

5-i 

5-i 

5-i 

6 

E.S.E. 

W.S.W. 

E.  by  N. 

W.  by  N. 

6-1 

6-1 

6-1 

7 

70  18  45 
73  7 30 
75  56  15 
78  45  0 

6-i 

6-i 

6-i 

7 

E.  by  S. 

W.  by  S. 

E. 

W. 

7-1 

7-1 

7-1 

8 

81  33  45 
84  22  30 
87  11  15 
90  0 0 

7-i 

7-i 

7-i 

8 

E. 

W. 

The  compass  varies  in  different  parts  of  the  earth ; hence,  in  sailing,  the 
compass  course  is  not  the  same  as  the  true  course.  The  true  course  is  the  com- 
pass course,  with  allowances  for  variation  of  the  needle  in  different  parts  of  the 
earth,  for  deviation  caused  by  the  iron  in  the  ship,  and  for  leeway,  the  angle 
which  the  ship  makes  with  her  track. 


PLANE  SAILING 


147 


Exercise  64.  Plane  Sailing 

1.  A ship  sails  from  latitude  40°  N.  on  a course  N.E.  26  mi.  Find 
the  difference  of  latitude  and  the  departure. 

2.  A ship  sails  from  latitude  35°  N.  on  a course  S.W.  53  mi.  Find 
the  difference  of  latitude  and  the  departure. 

3.  A ship  sails  from  a point  on  the  equator  on  a course  N.E.  by 
N.  62  mi.  Find  the  difference  of  latitude  and  the  departure. 

4.  A ship  sails  from  latitude  43°  45'  S.  on  a course  N.  by  E.  38  mi. 
Find  the  difference  of  latitude  and  the  departure. 

5.  A ship  sails  from  latitude  1°  45'  N.  on  a course  S.E.  by  E.  25  mi. 
Find  the  difference  of  latitude  and  the  departure. 

6.  A ship  sails  from  latitude  13°  17'  S.  on  a course  N.E.  by  E.  | E., 
until  the  departure  is  42  mi.  Find  the  difference  of  latitude  and  the 
latitude  reached. 

7.  A ship  sails  from  latitude  40°  20'  N.  on  a N.N.E.  course  for 
92  mi.  Find  the  departure. 

8.  If  a steamer  sails  S.W.  by  W.  20  mi.  what  is  the  departure 
and  the  difference  of  latitude  ? 

9.  If  a sailboat  sails  N.  25°  W.  until  the  departure  is  25  mi.,  what 
distance  does  it  sail  ? 

10.  A ship  sails  from  latitude  37°  40'  N.  on  a N.E.  by  E.  course 
for  122  mi.  Find  the  departure. 

11.  A yacht  sails  6^  points  west  of  north,  the  distance  being  12  mi. 
What  is  the  departure  ? 

12.  A steamer  sails  S.W.  by  W.  28  mi.  It  then  sails  N.W.  30  mi. 
How  far  is  it  then  to  the  west  of  its  starting  point  ? 

13.  A ship  sails  on  a course  between  S.  and  E.  24  mi.,  leaving 
latitude  2°  52'  S.  and  reaching  latitude  2°  58'  S.  Find  the  course  and 
the  departure. 

14.  A ship  sails  from  latitude  32°  18'  N.,  on  a course  between  N. 
and  W.,  a distance  of  34  mi.  and  a departure  of  10  mi.  Find  the 
course  and  the  latitude  reached. 

15.  A ship  sails  on  a course  between  S.  and  E.,  making  a differ- 
ence of  latitude  13  mi.  and  a departure  of  20  mi.  Find  the  distance 
and  the  course. 

16.  A ship  sails  on  a course  between  N.  and  W.,  making  a differ- 
ence of  latitude  17  mi.  and  a departure  of  22  mi.  Find  the  distance 
and  the  course. 


148 


PLANE  TRIGONOMETRY 


128.  Parallel  Sailing.  Sailing  dne  east  or  due  west,  remaining  on 
the  same  parallel  of  latitude,  is  called  parallel  sailing. 

129.  Finding  Difference  in  Longitude.  In  parallel  sailing  the  dis- 
tance sailed  is,  by  definition  (§  126),  the  departure.  Erom  the 
departure  the  difference  in  longitude  is  found  as  follows ; 

Let  be  the  departure.  Then  in  rt.  A OAD 

- lat. 

DA 

Hence  - = sin  (90°  — lat.)  = cos  lat 

(lA  ^ ' 


The  triangles  DAB  and  OEQ,  are  similar,  the  arcs  being  (§  125) 
considered  straight  lines. 


Therefore 


Hence 


Therefore 


^ _ dR  DA  _ AB 

oe~eq’  oa~eq' 

AB 

coslat  = — . 

AB 

EQ  = - AB  X sec  lat. 

cos  iat. 


That  is,  Diff.  long.  = depart,  x sec  lat. 

That  is,  the  number  of  minutes  in  the  difference  in  longitude  is  the  product 
of  the  number  of  miles  in  the  departure  by  the  secant  of  the  latitude,  the 
nautical,  or  geographic,  mile  being  a minute  of  longitude  on  the  equator. 


Exercise  65.  Parallel  Sailing 

1.  A ship  in  latitude  42°  16’  N.,  longitude  72°  16'  W.,  sails  due 
east  a distance  of  149  mi.  What  is  the  position  of  the  point  reached  ? 

2.  A ship  in  latitude  44°  49'  S.,  longitude  119°  42'  E.,  sails  due 
west  until  it  reaches  longitude  117°  16'  E.  Find  the  distance  made. 

3.  A ship  in  latitude  60°  15'  N.,  longitude  60°  15'  W.,  sails  due 
west  a distance  of  60  mb  What  is  the  position  of  the  point  reached  ? 


PLANE  SAILING 


149 


130.  Middle  Latitude  Sailing.  Since  a ship  rarely  sails  for  any 
length,  of  time  due  east  or  due  west,  the  difference  in  longitude  can- 
not ordinarily  be  found  as  in  parallel  sailing  (§  § 128, 129).  Therefore, 
in  plane  sailing  the  departure  between  two  places  is  measured  gen- 
erally on  that  parallel  of  latitude  which  lies  midway  between  the 


parallels  of  the  two  places.  This  is  called  the  method  of  middle 
latitude  sailing.  Hence,  in  middle  latitude  sailing, 

Diff.  long.  = depart,  x sec  mid.  lat. 

This  assumption  produces  no  great  error,  except  in  very  high  latitudes  or 
excessive  runs. 

Exercise  66.  Middle  Latitude  Sailing 

1.  A ship  leaves  latitude  31°  14'  N.,  longitude  42°  19'  W.,  and  sails 
E.N.E.  32  mi.  Find  the  position  reached. 

2.  Leaving  latitude  49°  57'  N.,  longitude  15°  16'  W.,  a ship  sails 
between  S.  and  W.  till  the  departm'e  is  38  mi.  and  the  latitude  is 
49°  38'  N.  Find  the  course,  distance,  and  longitude  reached. 

3.  Leaving  latitude  42°  30'  N.,  longitude  58°  51'  W.,  a ship  sails 
S.E.  by  S.  48  mi.  Find  the  position  reached. 

4.  Leaving  latitude  49°  57'  N.,  longitude  30°  W.,  a ship  sails 

S.  39°  W.  and  reaches  latitude  49°  44'  N.  Find  the  distance  and 
the  longitude  reached. 

5.  Leaving  latitude  37°  N.,  longitude  32°  16'  TV.,  a ship  sails  be- 
tween N.  and  W.  45  mi.  and  reaches  latitude  37°  10'  N.  Find  the 
course  and  the  longitude  reached. 

6.  A ship  sails  from  latitude  40°  28'  N.,  longitude  74°  W.,  on  an 
E.S.E.  course,  62  mi.  Find  the  latitude  and  longitude  reached. 

7.  A ship  sails  from  latitude  42°  20'  N.,  longitude  71°  4'  W.,  on  a 
N.N.E.  course,  30  mi.  Find  the  latitude  and  longitude  reached. 


150 


PLANE  TEIGONOMETRY 


131.  Traverse  Sailing.  In  case  a ship  sails  from  one  point  to  an- 
other on  two  or  more  different  courses,  the  departure  and  difference 
of  longitude  are  found  by  reckon- 
ing each  course  separately  and  com- 
bining the  results.  For  example, 
two  such  courses  are  shown  in  the 
figure.  This  is  called  the  method 
of  traverse  sailing. 

No  new  principles  are  involved  in 
traverse  sailing,  as  will  be  seen  in  solv- 
ing Ex.  1,  given  below. 

Exercise  67.  Traverse  Sailing 

1.  Leaving  latitude  37°  16'  S.,  longitude  18°  42'  W.,  a ship  sails 
N.E.  104  mi.,  then  N.N.W.  60  mi.,  then  W.  by  S.  216  mi.  Find  the 
position  reached,  and  its  bearing  and  distance  from  the  point  left. 

For  the  first  course  we  have  difierence  of  latitude  73.5  N.,  departure  73.5  E.; 
for  the  second  course,  difference  of  latitude  55.4  N.,  departure  23  W.;  for  the 
third  course,  difierence  of  latitude  42.1  S.,  departure  211.8  W. 

On  the  whole,  then,  the  ship  has  made  128.9  mi.  of  north  latitude  and  42.1  mi. 
of  south  latitude.  The  place  reached  is  therefore  on  a parallel  of  latitude  86.8  mi. 
to  the  north  of  the  parallel  left ; that  is,  in  latitude  35°  49.2'  S.  ^ 

In  the  same  way  the  departure  is  found  to  be  161.3  mi.  W.,  and  the  middle 
latitude  is  36°  32.6'.  With  these  data  we  find  the  difierence  of  longitude  to  be 
201',  or  3°  21'  W.  Hence  the  longitude  reached  is  22°  3'  W. 

With  the  difierence  of  latitude  86.8  mi.  and  the  departure  161.3  mi.,  we  find 
the  course  to  be  N.  61°  43'  W.  and  the  distance  183.2  mi.  The  ship  has  reached 
the  same  point  that  it  would  have  reached  if  it  had  sailed  directly  on  a course 
N.  61°  43'  W.  for  a distance  of  183.2  mi. 

2.  A skip  leaves  Cape  Cod  (42°  2'  N.,  70°  3'  W.)  and  sails  S.E.  by  S. 
114  mi.,  then  N.  by  E.  94  mi.,  then  W.N.AY.  42  mi.  Find  its  position 
and  the  total  distance. 

3.  A ship  leaves  Cape  of  Good  Hope  (34°  22'  S.,  18°  30'  E.)  and 
sails  N.W.  126  mi.,  then  N.  by  E.  84  mi.,  then  W.S.W.  21 T mi.  Find 
its  position  and  the  total  distance. 

4.  A ship  in  latitude  40°  N.  and  longitude  67°  4'  W.  sails  N.W. 

60  mi.,  then  N.  by  W.  52  mi.,  then  W.S.W.  83  mi.  Find  its  position. 

6.  A ship  sailed  S.S.W.  48  mi.,  then  S.W.  by  S.  36  mi.,  and  then  '. 
N.E.  24  mi.  Find  the  difference  in  latitude  and  the  departure. 

6.  A ship  sailed  S.  E.  18  mi.,  S.W.  ^ S.  37  mi.,  and  then  S.S.W 
I W.  56  mi.  Find  the  difference  in  latitude  and  the  departure. 


CHAPTER  X 


GRAPHS  OF  FUNCTIONS 


132.  Circular  Measure.  Besides  the  methods  of  measuring  angles 
which  have  been  discussed  already  and  are  generally  used  in 
practical  work,  there  is  another  method  that  is  frequently  employed 
in  the  theoretical  treatment  of  the  subject.  This  takes  for  the  unit 
the  angle  subtended  by  an  arc  which  is  equal  in  length  to  the  radius, 
and  is  known  as  circular  measure. 


133.  Radian.  An  angle  subtended  by  an  arc  equal  in  length  to  the 
radius  of  the  circle  is  called  a radian. 

The  term  "radian”  is  applied  to  both  the  angle  and 
arc.  In  the  annexed  figure  we  may  think  of  a radius 
bent  around  the  arc  AB  so  as  to  coincide  with  it.  Then  | 

AAOB  is  a radian. 

134.  Relation  of  the  Radian  to  Degree  Measure. 

The  number  of  radians  in  360°  is  equal  to  the 
number  of  times  the  length  of  the  radius  is  contained  in  the  length 
of  the  circumference.  It  is  proved  in  geometry  that  this  number  is 
2 7T  for  all  circles,  tt  being  equal  to  3.1416,  nearly.  Therefore  the 
radian  is  the  same  angle  in  all  circles. 

The  circumference  of  a circle  is  2 tt  times  the  radius. 

Hence  2 tt  radians  = 360°,  and  tt  radians  = 180°. 

1 80° 

Therefore  1 radian  = = 57.29578°  = 57°  17'  45", 


and 


1 degree  = radian  = 0.017453  radian. 

loO 


135.  Number  of  Radians  in  an  Angle.  Prom  the  definition  of  radian 
we  see  that  the  number  of  radians  in  an  angle  is  equal  to  the  length 
of  the  subtending  arc  divided  by  the  length  of  the  radius. 

Thus,  if  an  arc  is  6 in.  long  and  the  radius  of  the  circle  is  4 in.,  the  number 
of  radians  in  the  angle  subtended  by  the  arc  is  6 in.  ^ 4 in.,  or  1^. 

This  may  be  reduced  to  degrees  thus  : 

1 J X 57.29578°  = 85.94367°, 

or,  for  practical  purposes,  11  x 57.3°  = 85.9°  = 85°  54'. 

151 


152 


PLANE  TKIGONOMETKY 


136.  Reduction  of  Radians  and  Degrees.  Erom  the  values  found  in 
§ 134  the  following  methods  of  reduction  are  evident : 

To  reduce  radians  to  degrees,  multiply  57°  17'  45" , or  57.29578°, 
by  the  number  of  radians. 

To  reduce  degrees  to  radians,  multiply  0.017453  by  the  number 
of  degrees. 

These  rules  need  not  be  learned,  since  we  do  not  often  have  to  make  these 
reductions.  It  is  essential,  however,  to  know  clearly  the  significance  of  radian 
measure,  since  we  shall  often  use  it  hereafter.  In  solving  the  following  problems 
the  rules  may  be  consulted  as  necessary. 

In  particular  the  student  should  learn  the  following  : 

360°  = 2 7T  radians,  60°  = ^ w radians, 

180°  = 7T  radians,  30°  = ^ tt  radians, 

90°  = ^7T  radians,  15°  = w radians, 

45°  = 5 7t  radians,  22.5°  = ^ tt  radians. 

The  word  radians  is  usually  understood  without  being  written.  Thus  sin  1-n 
means  the  sine  of  2 tt  radians,  or  sin  360°  ; and  tan  tt  means  the  tangent  of 
^ TT  radians,  or  45°.  Also,  sin  2 means  the  sine  of  2 radians,  or  sin  114.59156°. 


Exercise  68.  Radians 

Express  the  following  in  radians : 


1.  270°. 

3.  56.25°. 

5.  196.5°. 

7. 

200°. 

< 2.  11.25°. 

> 4.  7.5°. 

1 6.  1440°. 

*~8. 

3000°. 

Express  the  following  in  degree  measure : 

9.  1\TT. 

11.  l^TT. 

•^3.  A^tt. 

2 4 

15. 

6 77. 

10.  l^TT. 

12.  l\ir. 

^^^4.  Stt. 

16. 

10  77. 

State  the  quadrant  in  which  the  following  angles  lie 

17.  |7T. 

19.  1|7T. 

21.  2.5  TT. 

23. 

1. 

^-18.  JTT. 

20.  IjTT. 

^22.  —3.4  77. 

<^4. 

— 2. 

Express  the  following  in  degrees  and  also  in  radians  : 

25.  f of  four  right  angles.  27.  f of  two  right  angles. 

26.  I of  four  right  angles.  28.  | of  one  right  angle. 

^..29.  What  decimal  part  of  a radian  is  1°?  1'? 

30.  How  many  minutes  in  a radian  ? How  many  seconds  ? 

31.  Express  in  radians  the  angle  of  an  equilateral  triangle. 

^32.  Over  what  part  of  a radian  does  the  minute  hand  of  a clock 
move  in  15  min.  ? 


GEAPHS  OF  FUNCTIOi^lS 


153 


137.  Functions  of  Small  Angles.  Let  A OP  be  any  acute  angle,  and 
let  X be  its  circular  measure.  Describe  a circle  of  unit  radius  about 
0 as  center  and  take  Z.AOP'  -=—/-AOP.  Draw  the  tangents  to 
the  circle  at  P and  P',  meeting  OA  in  T.  Then  we  see  that 


chord  PP'  < arc  PP' 

<PT+  P'T. 

Dividing  by  2,  MP  < arc  AP  < PT, 
or  sin x<x<  tan  x. 

X 


Dividing  by  Sin  x, 
Whence 


1<- 


-<  sec  X. 


sin  X 


. sin  X 

1> > cos  X. 


Therefore  the  value  of  between  cos  x and  1. 


If,  now,  the  angle  x is  constantly  diminished,  cos  x approaches 
the  value  1. 

sm  X 

Accordingly,  the  limit  of  ■ ^ > as  x approaches  0,  is  1. 

Hence  when  x denotes  the  circular  measure  of  an  angle  near  0°  we  may 
use  X instead  of  sin  x and  instead  of  tan  x. 


For  example,  required  to  find  the  sine  and  cosine  of  V. 
If  x is  the  circular  measure  of  1', 


2 7T  3.14159  + 
360  X 60  “ 10800 


0.00029088  +, 


the  next  figure  in  x being  8. 

Nowsinx  > Obut  <x;  hence  sin  1' lies  between  0 and  0.000290889. 
Again,  cos  1'  ==  Vl  - sin^l'  > Vl- (0.0003)2  > 0.9999999. 

Hence  cos  1'  = 0.9999999  +. 

But,  as  above,  sin  x~>  x cos  x. 

.-.  sinl'  > 0.000290888  x 0.9999999 

> 0.000290888  (1  - 0.0000001) 

> 0.000290888  - 0.0000000000290888 

> 0.000290887. 

Hence  sin  V lies  between  0.000290887  and  0.000290889 ; that  is. 
to  eight  places  of  decimals, 

sinl'=  0.00029088  +, 
the  next  figure  being  7 or  8. 


154 


PLANE  TRiaONOMETRY 


Y 

P' 

r 

X'  0 

A 

Y 

138.  Angles  having  the  Same  Sine.  If  we  let  Z.XOP  = x,  in  this 
figure,  and  let  P'  be  symmetric  to  P with  respect  to  the  axis  YY’,  we 
shall  have  ZA0P'  = 180°  — a:,  ovir  — x.  And 

since  - = sin  x = sin  (tt  — x)  we  see  that  x and 

IT  — X have  the  same  sine. 

Furthermore,  sin  x = sin  (360°  x),  and 

sin  (180°  — a;)  = sin  (360°  + 180°  — x).  That 
is,  we  may  increase  any  angle  by  360°  without 
changing  the  sine.  Hence  we  have  sina:  = sin(7i.  • 360°  + x),  and 
sin  (180°  — x)  = sin(?i  • 360°  + 180°  — x).  Using  circular  measure 
we  may  write  these  results  as  follows : 

sin  X = sin  (2  kir  + a;),  and  sin  (tt  — a-)  = sin  (2  ^ + 1 tt  — a;). 
These  may  be  simplified  still  more,  thus  : 

' sin  X = sin  [nw  + (—  l)"a;] 

where  n is  any  integer,  positive  or  negative. 


Thus  if  n = 0 we  have  sin  a:  = sin  (0  • tt  + (—  l)“x)  = sin  z ; if  n = 1 we  have 
sin  X = sin  (tt  — x) ; if  n = 2 we  have  sin  x = sin  (2  tt  + x) ; and  so  on. 

Since  the  sine  is  the  reciprocal  of  the  cosecant,  it  is  evident  that  x and 
mr  + (—  l)"x  have  the  same  cosecant. 

To  find  four  angles  whose  sine  is  0. 2-588,  we  see  by  the  tables  that  sin  1 5°= 0.2588. 
Hence  we  have  sin  15°  = sin  [mr  + (—  1)"  • 15°]  = sin  (tt  — 1-5°)  = sin  (2  tt  -f  15°) 
= sin(37T  — 15°);  and  so  on. 


Exercise  69.  Sines  and  Small  Angles 

1.  Find  four  angles  whose  sine  is  0.2756. 

2.  Find  six  angles  whose  sine  is  0.5000. 

3.  Find  eight  angles  having  the  same  sine  as  I ir. 

4.  Find  four  angles  having  the  same  cosecant  as  | tt. 

6.  Find  four  angles  having  the  same  cosecgint  as  0.1  tt. 

Given  Tt  = 3.141592653589,  compute  to  eleven  decimal  places : 

6.  cos  V.  7 . sin  1'.  8.  tan  1'.  9.  sin  2'. 

10.  From  the  results  of  Exs.  6 and  7,  and  by  the  aid  of  the  formula 
sin  2x  = 2 sin  x cos  x,  compute  sin  2',  carrying  the  multiplication  to 
six  decimal  places.  Compare  the  result  with  that  of  Ex.  9. 

11.  Compute  sinl°  to  four  decimal  places. 

o 

X X“ 

12.  From  the  formula  cos  x = 1 — 2 sin^  > show  that  cos  x > 1 — -w  • 


GRAPHS  OF  FUNCTIONS 


165 


139.  Angles  having  the  Same  Cosine.  If  we  let  Z.XOP  = x,  in 
this  figure,  and  let  P'  be  symmetric  to  P with  respect  to  the  axis 
XX',  we  shall  have  Z.XOP'  — 360°  — x,  or  — x, 
depending  on  whether  we  think  of  it  as  a 
positive  or  a negative  angle.  In  either  case 
. . b , 

its  cosme  is  - > the  same  as  cos  x. 
r 

In  either  case  cos  x = cos  {n  • 360°  — x'). 

In  general,  cos  x = cos  (2  wtt  ± x), 
where  n is  any  integer,  positive  or  negative. 


Thus  if  ji  = 0,  we  have  cosx  = cos(±  x);  if  ti  = 1,  we  have  cosz  = cos(27t±  x); 
if  n = 2,  we  have  cos  x = cos  (4  tt  ± x) ; and  so  on. 

Since  the  cosine  is  the  reciprocal  of  the  secant,  it  is  evident  that  x and  2n7r  ±x 
have  the  same  secant. 


140.  Angles  having  the  Same  Tangent.  Since  we  have  tan x = -> 

— a ° 

and  tan  (180°  x)  = — - > we  see  that  tan  x = tan  (180°  + x).  In 

general  we  may  say  that 

tan  x = tan  (2  kir  + x)  = tan  (2  k^r  + tt  + a:). 

This  may  be  written  more  simply  thus : 

tan  x = tan  (wtt  + x~), 

where  x is  any  integer,  positive  or  negative. 


Thus  if  we  have  tan  2(P  given,  we  know  that  nir  + 20°  has  the  same  tangent. 
Writing  both  in  degree  measure,  we  may  say  that  n • 180°  + 20°  has  the  same 
tangent.  If  n = 1,  we  have  200° ; if  n = 2,  we  have  380° ; if  n = 3,  we  have  560° ; 
and  so  on.  Furthermore,  if  n = — 1,  we  have  —160°;  and  so  on. 

Since  the  cotangent  is  the  reciprocal  of  the  tangent,  it  is  evident  that  x and 
KTT  + X have  the  same  cotangent. 


Exercise  70.  Angles  having  the  Same  Functions 

1.  Find  two  positive  angles  that  have  \ as  their  cosine. 

2.  Find  two  negative  angles  that  have  ^ as  their  cosine. 

3.  Find  four  angles  whose  cosine  is  the  same  as  the  cosine  of  25°- 

4.  Find  four  angles  that  have  2 as  their  secant. 

5.  Find  two  positive  angles  that  have  1 as  their  tangent. 

6.  Find  two  negative  angles  that  have  1 as  their  tangent. 

7.  Find  four  angles  that  have  V§  as  their  tangent. 

8.  Find  four  angles  that  have  Vs  as  their  cotangent. 

9.  Find  four  angles  that  have  0.5000  as  their  tangent. 

10.  Find  four  negative  angles  whose  cotangent  is  0.6000. 


156 


PLANE  TKIGONOMETRY 


141.  Inverse  Trigonometric  Functions.  If  y = 5iux,  then  x is  the 
angle  whose  sine  is  y.  This  is  expressed  by  the  symbols  x = sin~^  y, 
or  X = arc  sin  y. 

In  American  and  English  books  the  symbol  sin-i  y is  generally  used ; on  the 
continent  of  Europe  the  symbol  arc  sin  y is  the  one  that  is  met. 

The  symbol  sin~^  y is  read  ” the  inverse  sine  of  y,”  ” the  antisine 
of  y,”  or  " the  angle  whose  sine  is  y.”  The  symbol  arc  sin  y is  read 
" the  arc  whose  sine  is  y,”  or  " the  angle  whose  sine  is  y.” 

The  symbols  cos-i  a;,  tan-i  z,  cot-^  x,  and  so  on  are  similarly  used. 

The  symbol  sin-^y  must  not  be  confused  with  (sin  The  former  means 

the  angle  whose  sine  is  y ; the  latter  means  the  reciprocal  of  siny. 

We  have  seen  (§  138)  that  sin~^  0.5000  may  be  30°,  150°,  390°,  510°, 
and  so  on.  In  other  words,  there  are  many  values  for  sin~^  x ; that  is. 

Inverse  trigonometric  functions  are  many-valued. 

142.  Principal  Value  of  an  Inverse  Function.  The  smallest  positive 
value  of  an  inverse  function  is  called  its  'principal  value. 

For  example,  the  principal  value  of  sin~i  0.6000  is  30°;  the  principal  value 
of  cos-10.5000  is  60°;  the  principal  value  of  tan-i(— 1)  is  135°;  and  so  on. 


11.  Find  the  value  of  the  sine  of  the  angle  whose  cosine  is 
that  is,  the  value  of  sin(cos"^^). 

lind  the  values  of  the  following : 


Exercise  71.  Inverse  Functions 


12.  sin(cos“^  Vs).  13.  sin(tan-^l).  14.  cos(cot~^l). 

Prove  the  following  formtdas  ; 


GRAPHS  CP  PUHCTIOHS 


157 


Find  four  values  of  each  of  the  following : 

19.  tan“^  0.5774.  21.  sin“^  0.9613. 

20.  cot“^  0.6249.  22.  sin“^  0.3256. 

25.  Solve  the  equation  y — sin“^^. 

26.  Pind  the  value  of  sin(tan~^-|-  + tan“^-^). 

27.  If  sin“^a:  = 2 cos“^x,  find  the  value  of  x. 

Prove  the  folloiving  formidas : 

28.  cos  (sin~^  x)  — Vl  — P. 

29.  cos  (2  sin“^  cc)  = 1 — 2 a;^. 

30.  sin(sin~^a:)  = a:. 

31.  sin  (sin'^a:  + sin"^?/)  = x Vl  — y^  + y Vl  — 

32.  tan~^  2 + tan“^  4 = ^ tt. 

33.  2 tan“^x  = tan“^[2  a; : (1  — x^)]. 

34.  2 sin“^x  = sin~^(2  X Vl  — x^). 

35.  2 cos“^x  = cos“^(2  x^  — 1). 

36.  3 tan~^x  = tan“^[(3  x — x^)  : (1  — 3 x^)]. 

37.  sin~’-  Vx  : y = tan“^  Vx  : (yy  — x). 

38.  sin“^  V(x  — ?/)  ; (x  — z)  = tan~'  V(x  — y~)  : (yy  — z) 

39.  sin"^x  = sec“^(l : Vl  — x^). 

40.  2 sec~^x  = tan~^  [2  Vx^  — 1 ; (2  — 3p)\. 

41.  tan~^  \ + tan“^  i = i 

42.  tan“^^  + tan~^  J = tan~^  |. 

43.  sin“^f  + sin"^^  = sin“^|-|. 

44.  sin“^-^  VS2  4-  sin-^^  ViT  = J tt. 

45.  sec-^  f + sec-i  ||  = 75°  45'. 

46.  tan~^(2  + Vs)  — tan“^(2  — Vs)=;  sec~^2. 

47.  tan“^  J + tan~^  i + tan"^  | + tan~^  i = i 

48.  sin“^x  4-  sin“^  Vl  — x^  = ^ tt. 

49.  sin~^0.5  + sin”^  ^ V3  = sin“^l. 

50.  tan~^  ^ = tan~^  i + tan~^  |. 

51.  tan~^0.5  + tan“^0.2  + tan~^0.125  = \ ’tt. 

52.  tan~^l  + tan“^2  + tan"^3  = tt. 

53.  tan~^  f + tan~^  + tan“^  — \ir. 

54.  cos“^^  VlO  + sin"^  i VS  = \ 'ir. 


23.  eot~^  0.2756. 

24.  eos“^  0.9455. 


158 


PLANE  TRIGONOMETRY 


143.  Graph  of  a Function.  As  in  algebra,  so  in  trigonometry,  it  is 
possible  to  represent  a function  graphically.  Before  taking  up  the 
subject  of  graphs  in  trigonometry  a few  of  the  simpler  cases  from 
algebra  will  be  considered. 

Suppose,  for  example,  we  have  the  expression  3x  + 2.  Since  the 
value  of  this  expression  depends  upon  the  value  of  x,  it  is  called  a 
function  of  x.  This  fact  is  indicated  by  the  equation 

fix)  = 3 X + 2, 


read  " function  x = 3 x + 2.”  But  since  /(x)  is  not  so  easil}*  written 
as  a single  letter,  it  is  customary  to  replace  it  by  some  such  letter  as 
y,  writing  the  equation 

y = 3 X + 2. 


If  X = 0,  we  see  that  y = 2 ; if  x = 1, 
then  ?/  = 5 ; and  so  on.  We  may  form  a 
table  of  such  values,  thus  : 


X 

y 

X 

y 

0 

2 

0 

2 

1 

6 

- 1 

- 1 

2 

8 

- 2 

-4 

3 

11 

-3 

— I 

We  may  then  plot  the  points  (0,  2),  (1,  5),  (2,  8),  • • •,  1,  — 1)? 

(—  2,  — 4),  • • .,  as  in  § 77,  and  connect  them.  Then  we  have  the 
graph  of  the  function  3 x + 2. 


The  graph  shows  that  the  function,  y or  f(x),  changes  in  value  much  more 
rapidly  than  the  variable,  x.  It  also  shows  that  the  function  does  not  become 
negative  at  the  same  time  that  the  variable  does,  its  value  being  2 when  x = 0, 
and  ^ when  x = — J.  This  kind  of  function  in  which  x is  of  the  first  degree 
only  is  called  a linear  function  because  its  graph  is  a straight  line. 


Exercise  72 . Graphs 

Plot  the  graphs  of  the  following  functions : 


1.  2x. 

2.  \x. 

3.  — X. 

4.  X 4- 1. 


6.  X — 1. 

6.  2 X + 1. 

7.  3 — X. 

8.  4 — lx. 


9.  — 2 — X. 

10.  2 X + 3. 

11.  2x  — 3. 

12.  3 — 2x. 


13.  0.5  X + 1.5. 

14.  1.4  X — 2.3. 

15.  - if  X — 2^. 

16.  -259-X  + 3I. 


GRAPHS  OF  FUHCTIOHS 


159 


144.  Graph  of  a Quadratic  Function.  We  shall  now  consider  func- 
tions of  the  second  degree  in  the  variable.  Such  a function  is 


called  a quadratic  fimction. 

Taking  the  function  x — 2,  we 
write 

y X — 2. 

Preparing  a table  of  values,  as  on 
page  158,  we  have 


X 

y 

X 

y 

0 

-2 

0 

-2 

1 

0 

- 1 

-2 

2 

4 

-2 

0 

3 

10 

-3 

4 

4 

18 

-4 

10 

In  order  to  see  where  the  function  lies  between  y = — 2 and  y = — 2,  we 
let  X = — We  find  that  when  x = — 1,  y =—  2t.  Similarly  if  we  give  to  x 
other  values  between  0 and  — 1,  we  shall  find  that  y in  every  case  lies  between 
0 and  — 2. 


Plotting  the  points  and  drawing  through  them  a smooth  curve,  we 
have  the  graph  as  here  shown. 

This  curve  is  2,parabola.  All  graphs  of  functions  of  the  form  y = ox^  + 6x  + c 
are  parabolas.  

Graphs  of  functions  of  the  form  x^  + 2/^  = or  j/  = ± Vr^  — x^,  are  circles 
with  their  center  at  O. 

Graphs  of  functions  of  the  form  a^x*  + h^y-  — c^  are  ellipses,  these  becoming 
circles  if  a = 6. 

Graphs  of  functions  of  the  form  a^x^  — b-y"^  = are  hyperbolas. 

There  are  more  general  equations  of  all  these  conic  sections,  but  these  suffice 
for  our  present  purposes.  The  graph  of  every  quadratic  function  in  x and  y is 
always  a conic  section. 


Exercise  73.  Graphs  of  Quadratic  Functions 


Plot  the  graphs  of  the  following  functions  : 


1.  x^.  5.  x^  — 1.  9.  2cc^-f-3a:. 

2.  2x^.  6.  + a: -f- 1.  10.  3a;^  — 4x. 

3.  ix\ 


7.  x^  — x + 1.  11.  ± VT 


x . 


4.  + 1.  8.  + a:  — 1.  12.  ± V9  — 4 ; 


13.  ± V4  — 3 

14.  ± Vs  — 2 X*. 

15.  ± V4  + 3x^. 

16.  ± Vs  -h  2 a^. 


160 


PLANE  TEIGONOMETRA 


145.  Graph  of  the  Sine.  Since  sin  a;  is  a function  of  x,  we  can  plot 
the  graph  of  sin  x.  We  may  represent  x,  the  arc  (or  angle),  in  de- 
grees or  in  radians  on  the  x-axis.  Representing  it  in  degrees,  as 
more  familiar,  we  may  prepare  a table  of  values  as  follows : 


X — 

0° 

16° 

30° 

45° 

60° 

75°  90° 

105° 

120° 

135°  150° 

165° 

180°  . . . 

y = 

0 

.26 

.5 

.7 

.87 

.97  1 

.97 

.87 

.7  .5 

.26 

0 ... 

If  we  represent  each  unit  on  the  y-axis  by  and  each  unit  on  the 
x-axis  by  30°,  the  graph  is  as  follows : 


The  graph  shows  very  clearly  that  the  sine  of  an  angle  x is  positive  between 
the  values  x = 0°  and  x = 180°,  and  also  between  the  values  x = — 360°  and 
X = — 180°  ; that  it  is  negative  between  the  values  x = — 180°  and  x = 0°,  and 
also  between  the  values  x = 180°  and  x = 360°.  It  also  shows  that  the  sine 
changes  from  positive  to  negative  as  the  angle  increases  and  passes  through 
— 180°  and  180°,  and  that  the  sine  changes  from  negative  to  positive  as  the 
angle  increases  and  passes  through  the  values  — 360°,  0°,  and  360°.  These  facts 
have  been  found  analytically  (§84),  but  they  are  seen  more  clearly  by  studying 
the  graph. 

If  we  use  radian  measure  for  the  arc  (angle),  and  represent  each 
unit  on  the  a:-axis  by  0.1  vr,  the  graph  is  as  follows  : 


The  nature  of  the  curves  is  the  same,  the  only  difference  being  that  we  have 
used  different  units  of  measure  on  the  x-axis,  thus  elongating  the  curve  in  the 
second  figure. 

146.  Periodicity  of  Functions.  This  curve  shows  graphically  what 
we  have  already  found,  that  periodically  the  sine  comes  back  to  any 
given  value. 

Thus  sin  x = 1 when  x = — 270°,  90°,  450°,  • • •,  returning  to  this  value  for 
increase  of  the  angle  by  every  360°,  or  2 :r  radians.  The  period  of  the  sine  is 
therefore  said  to  be  360°  or  2 tt. 


GRAPHS  OF  FUNCTIONS 


161 


Exercise  74.  Graphs  of  Trigonometric  Functions 

1.  Verify  the  following  plot  of  the  graph  of  cos  x : 


rP 

X 

1 

1 

f 

1 

VI 

\ 

1 

r2 

IQ 

WftV 

9f 

* 

1 

3^60 

■ 

1 i 

< 

fin 

M7 

lb 

f 

1 / 

1 l\ 

ill) 

■*111 

2.  What  is  the  period  of  cos  x ? 

3.  Verify  the  following  plot  of  the  graph  of  tan  x : 


i-i  ' 1 ’ 

1 1 

! i 

i 

- 

■90-^ 

0 90^ 

* •>? 

* 

1 

1 ' 

/I 

7 

i 

1 

il  I 

;T 

4.  What  is  the  period  of  tan  x ? 

6.  Verify  the  following  plot  of  the  graph  of  cot  x : 


1 ■ . ' 1 ^ 

' 1 ■ 

; 1 

TW 

1 

V 

V 

V 

V 

-360 

Sii  * 

80*  *27( 

pla, 

S60 

■ 

: 

' 

' ' 

~r 

1 1 

6.  What  is  the  period  of  cot  x ? 

7.  Verify  the  following  plot  of  the  graph  of  sec  os: 


8.  What  is  the  period  of  sec  x ? 

9.  Plot  the  graph  of  esc  x,  and  state  the  period.  Also  state  at 
what  values  of  x the  sign  of  csccc  changes. 

10.  Plot  the  graphs  of  sin  x and  cos  x on  the  same  paper.  What 
does  the  figure  tell  as  to  the  mutual  relation  of  these  functions  ? 


162 


PLANE  TRIGOXOMETKY 


Exercise  75.  Miscellaneous  Exercise 

Find  the  areas  of  the  triangles  in  which : 

1.  a = 25,  = 25,  c = 25.  3.  a =74,  h = 75,  c = 92. 

2.  a = 25,  6 = 33J,  c = 41f . 4.  a = 2^,h  = 3^,  c = 4i. 

5.  Consider  the  area  of  a triangle  with  sides  17.2,  26.4,  43.6. 

6.  Consider  the  area  of  a triangle  with  sides  26.3,  42.4,  73.9. 

7.  Two  inaccessible  points  A and  B are  visible  from  D,  but  no 
other  point  can  be  found  from  which  both  points  are  visible.  Take 
some  point  C from  which  both  A and  D can  be  seen  and  measure  CD, 
200  ft. ; angle  ADC,  89°;  and  angle  A CD,  50°  30'.  Then  take  some 
point  E from  which  both  D and  B are  visible,  and  measure  DE, 
200  ft.;  angle  BDE,  54°  30';  and  angle  BED,  88°  30'.  At  D measure 
angle  ADB,  72°  30'.  Compute  the  distance  AB. 

8.  Show  by  aid  of  the  table  of  natural  sines  that  sin  x and  x agree 
to  four  places  of  decimals  for  all  angles  less  than  4°  40'. 

9.  If  the  values  of  log  x and  log  sin  x agree  to  five  decimal  places, 
find  from  the  table  the  greatest  value  x can  have. 

10.  Find  four  angles  whose  cosine  is  the  same  as  the  cosine  of  175°. 

11.  Find  four  angles  whose  cosine  is  the  same  as  the  cosine  of  200°. 

12.  How  many  radians  in  the  angle  subtended  by  an  arc  7.2  in. 
long,  the  radius  being  3.6  in.  ? How  many  degrees  ? 

13.  How  many  radians  in  the  angle  subtended  by  an  arc  1.62  in. 
long,  the  radius  being  4.86  in.  ? How  many  degrees  ? 

Draw  the  following  angles  : 

14.  — 7T.  16.  —^7T.  18.  2.7  7T.  20.  3 7T  — 9. 

16.  —2.  17.  — 19.  2 7t  — 6.  21.  4— 7T. 

22.  Find  four  angles  whose  tangent  is  • 

23.  Find  four  angles  whose  cotangent  is  • 

Vs 

24.  Plot  the  graphs  of  sin  x and  esc  x on  the  same  paper.  What 
does  the  figure  tell  as  to  the  mutual  relation  of  these  functions  ? 

26.  Plot  the  graphs  of  cos  x and  sec  x on  the  same  paper.  What 
does  the  figure  tell  as  to  the  mutual  relation  of  these  functions  ? 

26.  Plot  the  graphs  of  tan  x and  cot  x on  the  same  paper.  What 
does  the  figure  tell  as  to  the  mutual  relation  of  these  functions  ? 


CHAPTER  XI 


TRIGONOMETRIC  IDENTITIES  AND  EQUATIONS 


147.  Equation  and  Identity.  An  expression  of  equality  which,  is 
true  for  one  or  more  values  of  the  unknown  quantity  is  called  an 
equation.  An  expression  of  equality  which  is  true  for  all  values  of 
the  literal  quantities  is  called  an  identity. 

For  example,  in  algebra  we  may  have  the  equation 
4 a:  — 3 = 7, 


which  is  true  only  if  x = 2.5.  Or  we  may  have  the  identity 
(a  + 5)2  = a2  + 2ab  + b'^, 


which  is  true  whatever  values  we  may  give  to  a and  b. 

Thus  sin  x = 1 is  a trigonometric  equation.  It  is  true  for  x = 90°  or  ^ tt, 
X = 450°  or  2^7t,  X = 810°  or  41  tt,  and  so  on,  with  a period  of  360°  or  2 7t.  In 
general,  therefore,  the  equation  sin  x = 1 is  true  for  x = (2  n + ^)  tt.  It  is  this 
general  value  that  is  required  in  solving  a general  trigonometric  equation. 

On  the  other  hand,  the  equation  sin^x  = 1 — cos^x  is  true  for  all  values  of  x. 
It  is  therefore  an  identity. 

The  symbol  s is  often  used  instead  of  = to  indicate  identity,  but  the  sign  of 
equality  is  very  commonly  employed  unless  special  emphasis  is  to  be  laid  upon 
the  fact  that  the  relation  is  an  identity  instead  of  an  ordinary  equation. 


^ 148.  How  to  prove  an  Identity.  A convenient  method  of  proving 
a trigonometric  identity  is  to  substitute  the  proper  ratios  for  the 
functions  themselves. 

a c 

Thus  to  prove  that  sin  x = 1 : esc  x we  have  only  to  substitute  - for  sin  x and  - 
a c c a 

for  CSC  X.  We  then  see  that  Similarly,  to  prove  that  tan  x = sin  x sec  x, 

we  may  substitute  - for  tanx,  - for  sinx,  and  - for  seex.  We  then  have 
be  b 

a _a  c 
b c b 


We  can  often  prove  a trigonometric  identity  oy  reference  to 
formulas  already  proved. 


This  was  done  in  proving  the  identity  sin2x  = 2slnxcosx  (§  101),  and  in 
tanx  + tanw 

proving  tan(x  + y)  = (§  93). 

1 — tan  X tan  y 


In  some  cases  it  may  be  better  to  draw  a figure  and  use  a geometric 
proof,  as  was  done  in  § 90. 


163 


164 


PLANE  TRIGONOMETRY 


Exercise  76.  Identities 


Prove  the  following  identities  : 
2 tan  ^ X 


tan  X = 


2.  sin  a:  = 


3.  sin  2 X = 


1 — tan^  X 
2 tan  ^ X 
1 + tan^  \ X 
2 tan  X 


6.  tan  3 X = 


3 tan  X — tan*x 


1 + tan^x 


8. 


1—3  tan^x 
tan  2 X + tan  x _ sin  3 x 
tan  2 X — tan  x sin  x 
3 cos  X 4-  cos  3 X 


3 sin  X — sin  3 x 


= cot®x. 


4.  2sinx  + sin2x  = 
6.  sin  3 X = 


2 sin®x 
1 — cos  X 
sin^  2 X — sin^  x 


sin  3 X + sin  5 x 
9.  ^ r — = cot  X . 


sin  X 


10. 


11.  sin  X + sin  3 x + sin  5 x = 

12.  tan  2 X + sec  2 x = 

13.  tanx  + tany 

14.  tan  (x  + y)  = 


cos  3 X — cos  5 X 
sin  3 X 4-  sin  5 x 
sin  X 4-  sin  3 x 

sin^  3 X 


= 2 cos  2 X. 


sinx 
cos  X + sin  X 
cos  X — sin  X 

_ sin  (x  + y) 
cos  X cos  y 

sin  2x4-  sin  2 y 


16. 


cos  2x4-  cos  2 y 

sin  X + cos  y _ tan  fi  (^  + y)  + 
sin  X — cos  y tan  [-^  (x  — y)  — 45°] 


16.  sin  2 X 4-  sin  4 x = 2 sin  3 x cos  x. 

17.  sin  4 X = 4 sin  X cos  x — 8 sin^x  cos  x. 

18.  sin  4 X = 8 cos®x  sin  x — 4 cos  x sin  x. 

19.  cos  4 X = 1 — 8 cos^x  + 8 cos*x  =1  — 8 sin^x  + 8 sin'x. 

20.  cos  2 X + cos  4 X = 2 cos  3 X cos  x. 

21.  sin  3 X — sin  X = 2 cos  2 X sin  x. 

22.  sin^x  sin  3 X 4-  cos^x  cos  3 x = cos®  2 x 

23.  cos*x  — siiPx  = cos  2 x. 

24.  cos^x  + siiPx  = l — i sin^  2 x. 

25.  cos®x  — sin®x  = (1  — siiRx  cos®x)  cos  2 x. 

26.  cos®x  4-  sin®x  = 1 — 3 ,sin®x  cos®x. 

27.  CSC  X — 2 cot  2 X cos  X = 2 sin  X. 


IDENTITIES  AND  EQUATIONS 


165 


Prove  the  following  identities : 

28.  (sin  2x  — sin  2 y)  tan  (x  y)  = 2 (sin^x  — sin^i/). 

29.  sin  3 a;  = 4 sin  x sin  (60°  + x')  sin  (60°  — x'). 

30.  sin  4 £c  = 2 sin  x cos  3 a:  + sin  2 x. 

3 1.  sin  X + sin  (a:  — f tt)  + sin  (fir  — x)=  0. 

32.  cos  X sin  (y  — s)  + cos  y sin  (z  — x')-\-  cos  z sin  (x  — y~)=  0. 

33.  cos  (x  + y)  sin  y — cos  (x  + z)  sin  z 

= sin  (x  + y)  cos  y — sin  (x  + z)  cos  z. 

34.  cos  (a;  + y + s)  + cos  (x  y — z)-\-  cos  (x  — y z~) 

+ cos  (2/  + « — a;)  = 4 cos  x cos  y cos  z. 

35.  sin  (x  + y)  cos  (x  — y)-\-  sin  (7/  + z)  cos  {y  — z) 

+ sia  (z  + a;)  cos  (z  — x)=  sin  2 x -f-  sin  2y  sin  2 z. 

36.  sin  (x  + ?/)  + cos  (x  — y)  = 2 sin  (x  + ^ tt)  sin  (y  + i tt). 

37.  sin  (x  + y)—  cos  (x  — y)  = — 2 sin  (x  — ^ tt)  sin  {y  — ^ tt). 

38.  cos  (x  + y)  cos  (x  — y)=  cos^  x — sin^  y. 

39.  sin(x  + ?/)sin(x  — y)=  sin^x  — sin^y. 

40.  sin  X + 2 sin  3 x + sin  5 x = 4 cos^x  sin  3 x. 


If  A,  B,  C are  the  angles  of  a triangle,  prove  that : 

41.  sin  2 A 4-  sin  2B  + sin  2 C = 4 sin4  sin  A sin  C. 

42.  cos  2 A + cos  2B  + cos  2 C = — 1 — 4 cos4  cos£  cos  C. 

43.  sin  ZA  + sin  3A  + sin  3 C = — 4 cos  |4  cos  cos  | C. 

44.  cos^4  + cos^A  + cos^  (7  = 1 — 2 cos  A cos  A cos  C. 


If  A + B + C = 90°,  prove  that  : 


45.  tanA  tanA  + tan5  tan  C + tan  C tan  A = 1. 

46.  sin^A  + sin^A  + sin^  C = 1 — 2 sin  A sinA  sin  C. 

47.  sin  2A  + sin  2B  + sin  2(7  = 4 cos  A cos5  cosC. 

48.  Prove  that  cot~^  3 + csc“^  Vs  = J tt. 

49.  Prove  that  x + tan~^  (cot  2 x)  = tan~^  (cot  x). 


Prove  the  following  statements  : 

sin  76°  + sin  15°  ^ 

50.  . r— ^ = tan60°. 

sin  75  — sin  16 

51.  sin  60°  + sin  120°  = 2 sin  90°  cos  30°. 


62.  cos  20°  + cos  100°  + cos  140°  = 0. 

53.  cos  36°  + sin  36°  = V2  cos  9°. 

54.  tan  11°  15'  + 2 tan  22°  30'  + 4 tan  45°  = cot  11°  15'. 


166 


PLANE  TRIGONOMETRY 


149 . How  to  solve  a Trigonometric  Equation.  To  solve  a trigonometric 
equation  is  to  find  for  the  unknown  quantity  the  general  value  which 
satisfies  the  equation. 

Practically  it  suffices  to  find  the  values  between  0°  and  360°,  since  we  can 
then  apply  our  knowledge  of  the  periodicity  of  the  various  functions  to  give  us 
the  other  values  if  we  need  them. 

There  is  no  general  method  applicable  to  all  cases,  but  the  follow- 
ing suggestions  will  prove  of  value : 

1.  If  functions  of  the  sum  or  difference  of  two  angles  are  involved, 
reduce  such  functions  to  functions  of  a single  angle. 

Thus,  instead  of  leaving  sin  (x  y)  in  an  equation,  substitute  for  sin  (x  -t-  y) 
its  equal  sin  x cos  y + cosx  siny. 

Similarly,  replace  cos2x  by  cos^x  — sin^x,  and  replace  the  functions  of  ^x 
by  the  functions  of  x. 

2.  If  several  functions  are  involved,  reduce  them  to  the  same 
function. 

This  is  not  always  convenient,  but  it  is  frequently  possible  to  reduce  the 
equation  so  as  to  involve  only  sines  and  cosines,  or  tangents  and  cotangents, 
after  which  the  solution  can  be  seen. 

3.  If  possible,  employ  the  method  of  factoring  in  solving  the 
final  equation. 

4.  Check  the  results  by  substituting  in  the  given  equation. 

For  example,  solve  the  equation  cos  x = sin  2 x. 

By  § 101,  sin  2 X = 2 sin  x cos  x. 

.-.  cosx  = 2 sinx  cosx. 

(1  — 2 sin  x)  cos  x = 0. 

.-.  cosx  = 0,  orl  — 2sinx  = 0. 

.•.  X = 90°  or  270°,  30°  or  160°,  or  these  values  increased  by  2n?r. 

Each  of  these  values  satisfies  the  given  equation. 


Exercise  77.  Trigonometric  Equations 


Solve  the  following  equations: 

1.  sin  X = 2 sin  (■!■  tt  -|-  x). 

2.  sin  2 X = 2 cos  x. 

3.  cos  2 X = 2 sin  x. 

4.  sin  X + cos  X = 1. 

6.  sin  X + cos  2 X = 4 sin^x. 

6.  4 cos  2 X -h  3 cos  x = 1. 


iCh  cos  0 -f  cos  26  = 0. 

11.  cot  4 6 -f  CSC  6 = 2. 

12.  cot  X tan  2 X = 3. 


7.  sinx  = cos  2x. 

8.  tan X tan  2x  = 2. 

9.  secx  = 4 cscx. 


IDENTITIES  AND  EQUATIONS 


167 


Solve  the  following  equations : 


u 

13. 

sin  X + sin  2 x = sin  3 x. 

33. 

sin  X sec  2 x = 1. 

14. 

sin  2 X = 3 sin^x  — cos^x. 

34. 

sin^x  + sin  2 x = 1. 

15. 

cot  6 = ^ tan  6. 

35. 

cos  X sin  2 x esc  x = 1. 

u 

16. 

2 sin  6 = cos  6. 

36. 

cot  X tan  2 X = see  2 x. 

17. 

2 sin^x  + 5 sin  x = 3. 

37. 

sin  2 X = cos  4 x. 

18. 

tan  X sec  x = V2. 

38. 

sin  2 « cot  z — sin^  s = •§■• 

it' 

19. 

cos  X — cos  2 X = 1. 

39. 

tan^x  = sin  2 x. 

20. 

cos  3 X + 8 cos®x  = 0. 

40. 

sec  2 X + 1 = 2 cos  x. 

21. 

tan  X + cot  X = tan  2 x. 

41. 

tan  2 X + tan  3 x = 0. 

22. 

tan  X + sec  x = a. 

42. 

CSC  X = cot  X + Vs. 

23. 

cos  2 X = a (1  — cos  x). 

43. 

tan  X tan  3 x = — f . 

24. 

sin“^^x  = 30°. 

44. 

cos  5 X + cos  3 X + cos  x = 

0 

25. 

tan“^x  + 2 cot~^x  = 135°. 

45. 

sin^  X — cos^  X = k. 

26. 

sec  X — cot  X = CSC  x — tan  x. 

46. 

sin  X + 2 cos  x = 1. 

27. 

tan  2 X tan  x = 1. 

47. 

sin  4 X — cos  3 x = sin  2 x. 

28. 

tan'^x  + cot^x  = 

48. 

sin  X + cos  X = sec  x. 

29. 

sin  X + sin  2 x = 1 — cos  2 x. 

49. 

2 cos  X cos  3 X + 1 = 0. 

30. 

4 cos  2 X + 6 sin  x = 5. 

50. 

cos  3 X — 2 cos  2 X + cos  x = 

0. 

31. 

sin  4 X — sin  2 x = sin  x. 

51. 

sin  (x  — 30°)  = Vs  sin  x. 

32. 

2 sin^x  + sin^  2x  = 2. 

52. 

sin~^x  + 2 eos“^x  = f tt. 

53.  sin“^x  + 3 COS' 

■^x  = 

210°. 

64. 


tana: 


= cos  2 X. 


1 + tan  X 

55.  tan  ir  x) tan  tt  — x)  = 4. 

56.  Vl  -|-  sinx  — VI  — sin  X = 2 cos  x. 

57.  sin  (45°  + x)  + cos  (45°  — x)  = 1. 

58.  (1  — tan  x)  cos  2 x = a (1  + tan  x). 

59.  sin®x  + cos®x  = — ^ 

60.  sec  (x  + 120°)  + sec  (x  — 120°)  = 2 cos  x. 

61.  sin^x  cos^x  — cos^x  — sin^x  + 1 = 0. 

62.  sin  X + sin  2 x + sin  3 x = 0. 

63.  sin  6 + 2 sin  2 6 + 3 sin  36  = 0. 

64.  sin  3 X = cos  2 x — 1. 

66.  sin  (x  + 12°)  + sin  (x  — 8°)  = sin  20°. 


168 


PLANE  TRIGONOMETRY 


Solve  the  following  equations : 

66.  tan (60°  + a:)  tan (60°  — x)=~  2. 

67.  tan  x + tan  2 a:  = 0. 

68.  sin  (x  + 120°)  + sin  (x  + 60°)  = |. 

69.  sin  (x  + 30°)  sin  (x  — 30°)  = J. 

70.  sin  2 0 = cos  3 0. 

71.  sin*x  + cos^x  = |. 

72.  sin^a;  — cos^a:  = 

7 3.  tan  (x  + 30°)  = 2 cos  x. 

74.  sec  X = 2 tan  x + 

75.  sin  11a;  sin  4x  + sin  5 a:  sin  2x  = 0. 

76.  cos  X + cos  3 a:  + cos  6 a;  + cos  7 a:  = 0. 

77.  sin  (x  + 12°)  cos  (a;  — 12°)  = cos  33°  sin  57° 

78.  sin'^a:  + sin~^  ^ x =120°. 

79.  tan~^a:  + tan~^  2x  = tan~^  3 V3. 

80.  tan“^(a:  + 1)  + tan~^  (a:  — 1)  = tan"  ^ 2 a;. 

81.  (3  — 4 cos^a:)  sin  2 a:  = 0. 

82.  cos  2 0 sec  0 + sec  0 + 1 = 0. 

83.  sin  X cos  2 x tan  x cot  2 x sec  x esc  2x  =1. 

84.  tan  (0  -f  45°)  = 8 tan  0. 

85.  tan(0  + 45°)  tan  0 = 2. 

86.  sin  X + sin  3 a:  = cos  x — cos  3 x. 

87.  sin  \ X (cos  2 x — 2)  (1  — tan^x)  = 0. 

88.  tan  X + tan  2 x = tan  3 x. 

89.  cot  X — tan  X = sin  x + cos  x. 

Prove  the  following  identities : 

...  , , V / • s secx  esex 

90.  (1  4- cotx  4- tanx)  (sinx  — cosx)= — 5 

^ ^ csc'^x  sec'^x 

91.  2 CSC  2x  cotx  = 1 4- cot^x. 

92.  sin  a + sin  b + sin  (a  -\-h')=  1 cos  ^ a cos  ^ b sin  ^ (a  + i). 

93.  tan  (45°  + x)  — tan  (45°  — x)  = 2 tan  2 x. 

94.  cot^x  — cos^x  = cot**x  cos^x. 

95.  tan^x  — sin^x  = tan^x  sin^x. 

96.  cot^x  + cot^x  = csc^x  — CSC'^X. 

97.  cos^x  4-  sin^x  cos^y  = cos^y  4-  sin^^y  cos®x. 


IDENTITIES  AND  EQUATIONS 


169 


150.  Simultaneous  Equations.  Simultaneous  trigonometric  equations 
are  solved  by  the  same  principles  as  simultaneous  algebraic  equations. 

1.  Eequired  to  solve  for  x and  y the  system 


X sin  a -1-  y sin  b = m 

(1) 

X cos  a + y cos  b = n 

(2) 

From  (1), 

X sin  a cos  a + y sin  6 cos  a = m cos  a. 

(3) 

jj'rom  (2), 

X sin  a cos  a-\-  y cos  6 sin  a = n sin  a. 

(4) 

From  (3)  and  (4), 

y sin  b cos  a — y cos  6 sin  a = m cos  a — n sin  a. 

or 

y sin  {p  — a)  = m cos  a — n sin  a ; 

whence 

771  COS  a — n sin  a 

y — , 

sin  (6  — a) 

Similarly, 

n sin  b — m cos  b 

X = • 

sin  (6  — a) 

2.  Eequired  to  solve  for  x and  y the  system 

sin  X + sin  y = a 

(1) 

cos  X + cos  y = b 

(2) 

By  § 103, 

2 sin  \(x  + y)  cos  \ {x  — y)  = a, 

(3) 

and 

2 cos  ^ (x  + 2/)  cos  \ {x  — y)  = b. 

Dividing, 

tan  ^(x  + y)  = ^. 

(4) 

sin  h{x  + y)  = — 

Va^  + 

Substituting  the  value  of  sin  J (x  + j/)  in  (3), 

cos  ^{x  — y)=  J Va^  + 

(5) 

From  (4), 

X + 2/  = 2 tan- 1 - . 

b 

(6) 

From  (5), 

X — y = 2 cos- 1 J Va^  + 6*. 

(7) 

From  (6)  and  (7),  x = tan-  ^ ^ + cos-i  ^ Va^  + 

and 

y — tan-  ^ ^ ~ cos-  ^ ^ Va^  + 6^. 

3.  Eequired  to  solve  for  x and  y the  system 

y sin  X — a 

(1) 

y cos  X = b 

(2) 

Dividing, 

tan  X = - . 
h 

* 1 ® 

.-.  X = tan-i-. 

Adding  the  squares  of  (1)  and  (2), 

y‘‘  (sin^x  + cos^x)  = + b^. 

Therefore  + 6^, 

and  y = ± Va^  + 6^. 


170 


PLANE  TPIGONOMETPY 


4,  Eequired  to  solve  for  x and  y tlie  system 
y sin  (a:  + <z)  = m 
y cos  (a;  + ^)  = 

From  (1),  y sin  x cos  a + y cos  x sin  a = m. 

From  (2),  y cos  x cos  b — y sin  x sin  b = n. 

We  may  now  solve  for  y sinx  and  y cosx,  and  then  solve  for  x and  y. 


5.  Required  to  solve  for  r,  x,  and  y the  system 

r cos  X sin  y = a 
r cos  X cos  y — h 


Dividing  (1)  by  (2), 

Squaring  (1)  and  (2)  and  adding, 
Taking  the  square  root, 

Dividing  (3)  by  (5), 

Squaring  (3)  and  adding  to  (4), 


r Sin  x = c 

tan  y — 

^ b 

y = tan-i-- 
0 

cos^  X = + b^. 

r cos  X = V + 6^. 

, c 

tanx  = — — 

Va2  + 62 

V.  X = tan-i  — — ■ 

Va^  + 62 

r2  = + 6^  + c2. 

r = Va^  + 62  + c2. 


(1) 

(2) 


(1) 

(2) 

(3) 


(5) 


Exercise  78.  Simultaneous  Equations 


1.  sin  X + sin  y = sin  a 

cos  X + cos  y = 1 + cos  a 

2.  sin^a;  + sin^y  = a 
cos^a;  — cos^y  = h 

3.  sin  X — sin  y = 0.7038 
cos  X — cos  y = — 0.7245 

4.  X sin  21°  + y cos  44°  = 179.70 
X cos  21°  + y sin  44°  = 232.30 


sin^a:  y = m 

cos^x  + y = ?i 

6.  sinx  + siny  ==  1 
sinx  — siny  = 1 

7.  cos  X + cos  y = a 
cos  2 X + cos  2 y = b 

8.  sin  X + sin  y = 2 m sin  a 
cos  X + cos  y = 2n  cos  a 


Solve  the  following  systems  for  x and  y : 

5. 


9.  Find  two  angles,  x and  y,  knowing  that  the  sum  of  their  sines 
is  a and  the  sum  of  their  cosines  is  b. 


Solve  the  following  systems  for  r and  x: 

10.  X sin  X = 92.344  11.  r sin  (x  — 19°  18')  = 59.4034 

r cos  X = 205.309  r cos  (x  — 30°  54')  = 147.9347 


IDENTITIES  AND  EQUATIONS 


171 


151.  Additional  S3nnbols  and  Functions.  It  is  the  oustom  in  advanced 
trigonometry  and  in  higlier  mathematics  to  represent  angles  by  the 
Greek  letters,  and  this  custom  will  be  followed  in  the  rest  of  this 
work  where  it  seems  desirable. 


The  Greek  letters  most  commonly  used  for  this  purpose  are  as  follows : 


a,  alpha 
/3,  beta 
7,  gamma 
5,  delta 
e,  epsilon 


6,  theta 
X,  lambda 
/i,  mu 
0,  phi 
omega 


Besides  the  six  trigonometric  functions  already  studied,  there  are, 
as  mentioned  on  page  4,  two  others  that  were  formerly  used  and 
that  are  still  occasionally  found  in  books  on  trigonometry.  These 
two  functions  are  as  follows  : 


versed  sine  of  or  = 1 — cos  a,  written  versin  a ; 
coversed  sine  of  or  1 — sin  a,  written  coversin  a. 


Exercise  79.  Simultaneous  Equations 


1.  Solve  for  and  x : 
versin  <f>  — x 

1 — sin<^  ---  0.5 

2.  Solve  for  6 and  x : 
1 — sin  6 = x 

1 + sin  6 = a 

3.  Solve  for  X and  /x : 
sin  X = V2  sin  ju, 
tan  X = Vs  tan  /a 


4.  Solve  for  0 and  : 
sin  0 + cos  (f)  = a 
sin  <f>  + cos  6 = b 

5.  Solve  for  0 and  : 

a siid0  — b sm*cj>  = a 
a cos^  6 — b cos^  <f)  — b 

6.  Solve  for  6 : 
sin^0  + 2 cos  6=2 
cos  6 — eos^d  - 0 


152.  Eliminant.  The  equation  resulting  from  the  elimination  of 
a certain  letter,  or  of  certain  letters,  between  two  or  more  given 
equations  is  called  the  eliminant  of  the  given  equations  with  respect 
to  the  letter  or  letters. 


For  example,  it  ax  = h and  a'x  = b',  it  follows  by  division  that  a : a'  = b : b', 
or  that  db'  — al),  and  this  equality,  in  which  x does  not  appear,  is  the  eliminant 
of  the  given  equations  with  respect  to  x. 

There  is  no  definite  rule  for  discovering  the  eliminant  in  trigo- 
nometric equations.  The  study  of  a few  examples  and  the  recalling 
of  identities  already  considered  will  assist  in  the  solutions  of  the 
problems  that  arise. 


172 


PLANE  TRIGONOMETRY 


153.  Illustrative  Examples.  The  following  examples  will  serve  to 
illustrate  the  method  of  finding  the  eliminant : 

1.  Find  the  eliminant,  with  respect  to  <j),  of 

sin<^  = a 
cos  <f>  = b 

Since  sin^^  + cos*0  = 1,  we  have  -{■  = 1,  the  eliminant. 

2.  Find  the  eliminant,  with  respect  to  X,  of 

sec  X = m 
tan  X — n 

Since  sec^X  — tan^X  = 1,  we  have  m?‘  — rfi  = 1,  the  eliminant. 

3.  Find  the  eliminant,  with  respect  to  ft.,  of 

m sin  ft.  + cos  ju.  = 1 
n sin  ft.  — cos  ft.  = 1 

Writing  the  equations  m sin  /*  = 1 — cos  ft,  n sin  = 1 + cos  /x,  and  multiplying, 
we  have 

mn  sin^yu  = 1 — cos^/x  = sin^yix. 

Hence  mn  = 1 is  the  eliminant. 


Exercise  80.  Elimination 

Find  the  eliminant  with  respect  to  a,  6,  X,  fi,  or  <f>  of  the  folloiv- 
ing  equations : 


1.  sin  1 = a 
cos  (j)  — 1 = b 

2.  tan  A — 0.  = 0 
cot  X — b = 0 

3.  sin  a m = n 
cos  a p = q 

4.  a + sec  <f>  = b 
p cot  <l>  = q 

5.  c sin  2<j)  + cos  2 <^  = 1 
b sin  2 <^  — cos  2 <f>=l 

6.  X = r(0  — sin  0) 
y = r(l  — cos  6) 

9 = versine- 1 y/r. 


7.  sin  </)  + sin  2 ^ = m. 
cos  (f>  + cos  2 <f)  = n 

8.  a + sin  0 = esc  6 
b + cos  6 — sec  6 

9.  tan  a + ffin  a = m 
tan  a — sin  a = n 

10.  p sin^  — P cos*  p.  = r 

p'  cos*  ft.  —p'  sin*  fx  — r' 

11.  sin  2 <f>  + tan  2 <f>  = k 
sin  2 (f>  — tan  2 4>  = I 

12.  p = a cos  0 cos  <f> 
q = b cos  6 sin  <ft 
r = c sin  9 


CHAPTER  XII 


APPLICATIONS  OF  TRIGONOMETRY  TO  ALGEBRA 

154.  Extent  of  Applications.  Trigonometry  has  numerous  applica- 
tions to  algebra,  particularly  in  the  approximate  solutions  of  equations 
and  in  the  interpretation  of  imaginary  roots. 

These  applications,  however,  are  not  essential  to  the  study  of  spherical  trigo- 
nometry, and  hence  this  chapter  may  be  omitted  without  interfering  with  the 
student’s  progress. 

For  example,  if  we  had  no  better  method  of  solving  quadratic  equa- 
tions we  could  proceed  by  trigonometry,  and  in  some  cases  it  is  even 
now  advantageous  to  do  so.  Consider  the  equation  px  — q = Q, 
Here  the  roots  are 


= - iP  + h + 4 2,  x^  = — \p  — \ -1-4  2', 


ajj  = — cot  -t-  Vy  VcoP^  -t-  1 


= — cot  -1-  — cot 

sin  <f>  ^ Vsin  <j!>  / 


Similarly, 


= — -y/q  cot  J 4>. 

For  example,  if  -f  1.1102  x — 3.3594  = 0 we  have 


whence 


log  tan  0 = 0.51876, 

<j>  = 73°  9'  2.6". 
log  tan  \<t>  = 9.87041  —10. 


and 


Therefore 


and 


and 


Hence 


log  Vq  = logV3.3594  = 0.26313. 

logXj  = 0.13354, 
Xj  = 1.360. 


X2=- 2.470. 
173 


Similarly, 


174 


PLANE  TKIGONOMETEY 


155.  De  Moivre’s  Theorem.  Expressions  of  the  form 
cos  X + i sin  X, 

where  i = V—  1,  play  an  important  part  in  modern  analysis. 
Since  (cos  x i sin  x)  (cos  y i sin  y) 


= cos  X cos  y — sin  a;  sin  y + i (cos  cc  sin  y + sin  x cos  y') 
= cos  (x  + y)  + i sin  (x  + y), 
we  have  (cos  x + i sin x^  = cos  2x  + i sin  2 x ; 

and  again,  (cos  x-\-  i sin  (cos  x i sin  (cos  x i sin  x) 

= (cos  2 X + i sin  2 x)  (cos  x + i sin  x) 

= cos  3 X + i sin  3 x. 

Similarly,  (cos  x i sin  x)"=  cos  nx  i sin  nx. 


To  find  the  nth  power  of  cos  x-\-i  sin  x,  n being  a positive  integer, 
we  have  only  to  multiply  the  angle  x hy  n in  the  expression. 

This  is  known  as  De  Moivre’s  Theorem,  from  the  discoverer  (c.  1725). 


156.  De  Moivre’s  Theorem  extended.  Again,  if  n is  a positive  integer 
as  before. 


i X ..  x\ 
cos  - + t sin  - I = cos  X + i sin  x. 


, . . .-r  X . . X 

.'.  (cos  X 1 sin  x)  = cos  - + i sin  - • 

^ ' n n 

However,  x may  be  increased  by  any  integral  multiple  of  2 tt  with- 
out changing  the  value  of  cos  x -f  i sin  x.  Therefore  the  following  n 
expressions  are  the  «th  roots  of  cos  x i sin  x : 

X...X  x-|-27r,..x-}-27r 

cos  — \- 1 sin  - 5 cos h t sin > 

n n n n 

x + Itt  . . x-flvT 

cos h % sin > • ■ • j 


cos 


-f(?i  — 1)2  7t  , . . x-f(?i  — 1)2  7t 
-f  i sin 


Hence,  if  is  a positive  integer. 


(cos  X -t-  i sin  x)" 

X + 2 kir 


= cos 


, . . x + 2Jc7r  n 1 o -.N 

-f  tsm (*  = 0,1,  2,  . • •,re— 1). 


Similarly,  it  may  be  shown  that 

— Tft 

(cos  x-\-i  sin  xfi  = cos— (x  -f  2 *7t)  -|-  i sin— (x  -f  2 kif). 

(*  = 0, 1, 2,  • • •,  w — 1,  m and  n being  integers,  positive  or  negative.) 


APPLICATIONS  TO  ALGEBEA 


175 


157.  The  Roots  of  Unity.  If  we  have  the  binomial  equation 


x”  — 1=  0, 

we  see  that  x’‘  — 1, 

and  X = the  nth  root  of  1, 

of  which  the  simplest  positive  root  is  "VI  or  1.  Since  the  equation 

is  of  the  nth  degree,  there  are  n roots.  In  other  words,  1 has  n nth 

roots.  These  are  easily  found  by  De  Moivre’s  Theorem. 


There  are  no  other  roots  than  those  in  § 156.  For,  letting  k = n,  n + 1,  and  so 
on,  we  have 

X + n(2tr)  . . X + n(2Tr) 

cos t sm — — - 

n n 

(X  \ /x  \ X X 

- + 2 7T I + i sin  ( — + 2 7T ) = cos  - + i sin  — , 

n / \n  / n n 


and 


x + (n+l)27T  . . ®+(n+l)27r 

cos h i sin 


h 

= cos  I 


’x  + 2 7T 


n 

X + 2 7T 


+ 2 7T 


+ ^ sin 


■^  + i sin 

X + 27t 


'x  + 2 7T 


+ 2 


n n 

and  so  on,  all  of  which  we  found  when  & = 0,  1,  2,  • • • , n — 1. 

Por  example,  required  to  find  the  three  cube  roots  of  1. 
If  COS0  + i sin  0=1,  the  given  number, 

then  0 = 0,  2 7T,  4 7T,  • • • . 

Also  (cos0  + i sin0)’^  = 1^  = the  three  cube  roots  of  1. 

k(2Tr)  + <b  . . A:  (2  7t)  + 0 

But  (cos0  + ^ sin  0)3  z=  cos-= 1-  i sin — i ^ 

3 3 


where  fc  = 0,  1,  or  2,  and  0 = 0,  2 ir,  4 w,  • • • . 

Therefore  1^  = cos  2ir  + i sin  2 tt  = 1, 

or  l'^  = cos  + i sin  §7T  = cos  120°  + i sin  120° 

= _ ^ ^ Vs  . i =_  0.5  + 0.8660 i, 

or  1^  = cos  ^TT  + i sin  f w = cos  240°  + i sin  240° 

^ V3-  i =-  0.5-  0.86601. 

The  three  cube  roots  of  1 are  therefore 

1,  — ^ ^ y/  — 3,  — i y/—~S. 

These  roots  could,  of  course,  be  obtained  algebraically,  thus : 
x«  - 1 = 0, 

whence  (x  — 1)  (x^  + x + 1)  = 0 ; 

and  either  x — 1 = 0,  whence  x = 1, 

or  X*  + X + 1 = 0,  whence  x = — ^ ± i V — 3. 

Most  equations  like  x"  — a = 0 caimot,  however,  be  solved  algebraically. 


176 


PLANE  TRIGONOMETKY 


Required  to  find  the  seven  7th  roots  of  —1;  that  is,  to  solve  the 
equation  = — 1,  or  x’  + 1 = 0. 

If  COS0  + i sin0  = — 1,  the  given  number, 

then  0 = 7T,  3 7T,  5 7T,  • • • . 

Ai  / fc(2  7r)+0  . . . k(2Tr)  + d> 

Also  (COS0  + ^ sin0)7  = cos  — ^ ^ + i sm  — ^ ^ - , 

where  A:  = 0,  1,  • • • , 6,  and  0 = ir,  3 tt,  • • • . 

That  is,  in  this  case 

I ^ 1 • • (2fc  + l)7T  . . (2fc  + l)7T 

(cos0  + I sin0)t  = cos^^ — h i sm  ^ — - — 

Hence  the  seven  7th  roots  of  1 are 

cos^  + i siuy  = cos  25°  42'  51^"  + i sin  25*  42'  51^", 

3 7T  3 7T 

cos h i sin  — = cos  77°  8'  34t"  + i sin  77°  8'  342", 

7 7 ^ 

57t  . . 5tt  . . 9ir  . . 9tt 

cos h i sm  — , cos  7T  + ^ sm  tt,  cos  — • + i sm  — , 

7 7 7 7 

Htt  . . IItt  13-7r  . . 13  7T 

cos H i sm , cos h i sm 

7 7 7 7 

All  these  values  may  be  found  from  the  tables.  For  example, 

cos  25°  42'  51f"  + i sin  25°  42'  51^"  = 0.9010  + 0.4339  V^. 


and 


Exercise  81.  Roots  of  Unity 

1.  Find  by  De  Moivre’s  Theorem  the  two  square  roots  of  1. 

2.  Find  by  De  Moivre’s  Theorem  the  four  4th  roots  of  1. 

3.  Find  three  of  the  nine  9th  roots  of  1. 

4.  Find  the  five  5th  roots  of  1. 

5.  Find  the  six  6th  roots  of  + 1 and  of  — 1. 

6.  Find  the  four  4th  roots  of  — 1. 

7.  Show  that  the  sum  of  the  three  cube  roots  of  1 is  zero. 

8.  Show  that  the  sum  of  the  five  5th  roots  of  1 is  zero. 

9.  From  Exs.  7 and  8 infer  the  law  as  to  the  sum  of  the  nth 
roots  of  1 and  prove  this  law. 

10.  From  Ex.  9 infer  the  law  as  to  the  sum  of  the  nth  roots  of  k 
and  prove  this  law. 

11.  Show  that  any  power  of  any  one  of  the  three  cube  roots  of  1 
is  one  of  these  three  roots. 

12.  Investigate  the  law  implied  in  the  statement  of  Ex.  11  for  the 
four  4th.  roots  and  the  five  5th  roots  of  1. 


APPLICATIONS  TO  ALGEBEA 


177 


158.  Roots  of  Numbers.  We  have  seen  that  the  three  cube  roots 
oflare  ^ gog  120°  + i sin  120°  = - ^ ^ V^, 

= cos  240°  + i sin  240°  = — ^V—  3, 
and  = cos  360°  + i sin  360°  = cos  0°  + i sin  0°  = 1. 


Furthermore,  x^  is  the  square  of  x^,  because 

(cos  120°  4-  i sin  120°)^  = cos  (2  • 120°)  + i sin  (2  • 120°), 
by  De  Moivre’s  Theorem.  We  may  therefore  represent  the  three 
cube  roots  by  to,  a?,  and  either  to^  or  1. 

In  the  same  way  we  may  represent  all  n of  the  Tith  roots  of  1 by 


If  we  have  to  extract  the  three  cube  roots  of  8 we  can  see  at  once 

that  they  are  j o 2 

2j  2 CO5  and  2 <o , 


because  2®  = 8,  (2  to)^  = 2^  to^  = 8 • 1 = 8, 

and  (2  = 2"  to«  = 2®  (to®)"  2®  1"  = 8. 


In  general,  to  find  the  three  cube  roots  of  any  number  we  may 
bake  the  arithmetical  cube  root  for  one  of  them  and  multiply  this 
by  to  for  the  second  and  by  to"  for  the  third. 

The  same  is  true  for  any  root.  For  example,  if  w,  (o'*,  and  to®  or  1 are 

the  five  5th  roots  of  1,  the  five  5th  roots  of  32  are  2 to,  2 2 co^,  2 and  2 to®  or  2. 


Exercise  82.  Roots  of  Numbers 

1.  Find  the  three  cube  roots  of  125. 

2.  Find  the  four  4th  roots  of  — 81  and  verify  the  results. 

3.  Find  three  of  the  6th  roots  of  729  and  verify  the  results. 

4.  Find  three  of  the  10th  roots  of  1024  and  verify  the  results. 

5.  Find  three  of  the  100th  roots  of  1. 

6.  Show  that,  if  2 w is  one  of  the  complex  7th  roots  of  128,  two  of 
the  other  roots  are  2 to"  and  2 to®. 

7.  Show  that  either  of  the  two  complex  cube  roots  of  1 is  at  the 
same  time  the  square  and  the  square  root  of  the  other. 

8.  Show  that  a result  similar  to  the  one  stated  in  Ex.  7 can  be 
found  with  respect  to  the  four  4th  roots  of  1. 

9.  Show  that  the  sum  of  all  the  rath  roots  of  1 is  zero. 

10.  Show  that  the  sum  of  the  products  of  all  the  rath  roots  of  1, 
taken  two  by  two,  is  zero. 


178 


PLANE  TEIGONOMETEY 


159.  Properties  of  Logarithms.  The  properties  of  logarithms  have 
already  been  studied  in  Chapter  III.  These  properties  hold  true 
whatever  base  is  taken.  They  are  as  follows  : 

1.  The  logarithm  of  1 is  0. 

2.  The  logarithm  of  the  base  itself  is  1. 

3.  The  logarithm  of  the  reciprocal  of  a positive  number  is  the 
negative  of  the  logarithm  of  the  number. 

4.  The  logarithm  of  the  product  of  two  or  more  positive  numbers  is 
found  by  adding  the  logarithms  of  the  several  factors. 

5.  The  logarithm  of  the  quotient  of  two  positive  numbers  is  found 
by  subtracting  the  logarithm  of  the  divisor  from  the  logarithm  of  the 
dividend. 

6.  The  logarithm  of  a power  of  a positive  number  is  found  by 
multiplying  the  logarithm  of  the  number  by  the  exponent  of  the  power. 

7.  The  logarithm  of  the  real  positive  value  of  a root  of  a positive 
number  is  found  by  dividing  the  logarithm  of  the  number  by  the  index 
of  the  root. 

160.  Two  Important  Systems.  Although  the  number  of  different 
systems  of  logarithms  is  unlimited,  there  are  but  two  systems  which 
are  in  common  use.  These  are 

1.  The  common  system,  also  called  the  Briggs,  denary,  or  decimal 
system,  of  which  the  base  is  10. 

2.  The  natural  system,  of  which  the  base  is  the  fixed  value  which 
the  sum  of  the  series 

11  1 1 

2'^!. 2-3'^  1-2. 3. 

approaches  as  the  number  of  terms  is  indefinitely  increased.  This 
base,  correct  to  seven  places  of  decimals,  is  2.7182818,  and  is  denoted 
by  the  letter  e. 

Instead  of  writing  1 . 2,  1 • 2 • 3,  1 ■ 2 • 3 • 4,  and  so  on,  we  may  write  either 
2 !,  3 !,  4 !,  and  so  on,  or  [2,  [3,  |^,  and  so  on.  The  expression  2 ! is  used  on  the 
continent  of  Europe,  [2  being  formerly  used  in  America  and  England.  At  pres- 
ent the  expression  2 ! is  coming  to  he  preferred  to  [2  in  these  two  countries. 

The  common  system  of  logarithms  is  used  in  actual  calculation; 
the  natural  system  is  used  in  higher  mathematics. 

The  natural  logarithms  are  also  known  as  Naperian  logarithms,  in 
honor  of  the  inventor  of  logarithms,  John  Napier  (1614),  although 
these  are  not  the  ones  used  by  him.  They  are  also  known  as  hyper- 
bolic logarithms. 


APPLICATIONS  TO  ALGEBRA 


179 


161.  Exponential  Series.  By  the  binomial  theorem  we  may  expand 


and  have 


1+-  =l+x+ 

n/ 


21 


^ 3! 


+ 


(1) 


This  is  true  for  all  values  of  x and  n,  provided  n > 1.  If  n is  not  greater 
than  1 the  series  is  not  convergent ; that  is,  the  sum  approaches  no  definite  limit. 
The  further  discussion  of  convergency  belongs  to  the  domain  of  algebra. 


When  a:  = 1 we  have  i 

1-i 


1 + ^ 1 = 1 + 1 + ■ 


2! 


+ 


3! 


+ 


(2) 


But 


Hence,  from  (1)  and  (2), 


1-  - 

1 + 1 + -^  + 


2! 


= 1 + a:  + ■ 


3! 


-) 

A + 


(*-J) 

2!  3! 


(3) 


If  we  take  n infinitely  large,  (3)  becomes 

(l  + 1 + ...)=  1 + a; +fj + ... ; 

that  is,  e*  = 1 + a;  + — 4-  ^ 4-  • • •. 

In  particular,  if  a;  = 1 we  have 

e = 1 4- 1 4- 4- 4-  • • •. 


(4) 


We  therefore  see  that  we  can  compute  the  value  of  e 
by  simply  adding  1,  1,  ^ of  1,  of  of  1,  and  so  on, 
indefinitely,  and  that  to  compute  the  value  to  only  a few 
decimal  places  is  a very  simple  matter.  We  have  merely 
to  proceed  as  here  shown. 

Here  we  take  1, 1,  of  1,  of  ^ of  1,  i of  ^ of  ^ of  1, 
and  so  on,  and  add  them.  The  result  given  is  correct 
to  five  decimal  places.  The  result  to  ten  decimal  places 
is  2.7182818284. 


1.000000 
2 1.000000 

3 0.500000 

4 0.166667 

5 0.041667 

6 0.008333 

7 0.001388 

8 0.000198 

9 0.000025 
0.000003 

e = 2.71828. 


180 


PLANE  TEIGONOMETKY 


162.  Expansion  of  sin  x,  cos  x,  and  tan  x.  Denote  one  radian  by  1, 
and  let 

cos  l-\-  i sin  1 — k. 

Then  cos  x i sin  x = (cos  1 + i sin  ly  = 

and,  putting  — x for  x, 

cos  (—  x)-{-  i sin(—  x)  — cos  x — i sinx  = 

That  is,  cos  x i sin  x — k^, 

and  cos  x — i sin  x = kr"^. 

By  taking  the  sum  and  difference  of  these  two  equations,  and 
dividing  the  sum  by  2 and  the  difference  by  2 i,  we  have 

cos  x.  — \.  (7c*  + 7:“*), 

and  sin  x = ^.  (k^  — k~^). 

Jd  % 


But  /fc*  = (e’og*)*  = and  kr^  = 

(log  a;^(log  kY 


and 


and 


•.  = 1 + X log  k + 

g— “clog*:  _ 


2! 


3! 


+ 


cosx 


= I (4.  + ^- .) = 1 + + 

1 


sin  X = - -<  X log  k + 


x®(log7;)®  x®(log7:)® 

^ 


Dividing  the  last  equation  by  x,  we  have 


smx 


= - log  7:  + • 


(log  ky  ^ x^  (log  ky 


3! 


5! 


+ 


But  remembering  that  x represents  radians,  it  is  evident  that  the 
smaller  x is,  the  nearer  sin  x comes  to  equaling  x ; that  is,  the  more 
nearly  the  sine  equals  the  arc. 

Sin  X 

Therefore  the  smaller  x becomes,  the  nearer  comes  to  1,  and 


the  nearer  the  second  member  of  the  equation  comes  to  t log  k. 

We  therefore  say  that,  as  x approaches  the  limit  0,  the  limits  of 
these  two  members  are  equal,  and 


1 = T log  7: ; 


whence 

and 


log  k = i, 
k = e*'. 


APPLICATIONS  TO  ALGEBEA 


181 


Therefore,  we  have 


1 » I -Xi\  1 ^ , 

cosa;=-(e«  + e “)  = 1 “ ^ + 4]  “ ^ + 


1 . a:*'  X"  X'  , 

smx  = -(e“-6-“)  = x + 


„5  ^ 

^ 3!  ' 5!  7! 

From  the  last  two  series  we  obtain,  by  division. 


tan  X = 


since 
cos  X 


2x®  . 17 cc’ 


"^+3+15+315 


By  the  aid  of  these  series,  which  rapidly  converge,  the  trigonometric  func- 
tions of  any  angle  are  readily  calculated. 

In  the  computation  it  must  be  remembered  that  x is  the  circular  measure  of 
the  given  angle. 

Thus  to  compute  cosl,  that  is,  the  cosine  of  1 radian  or  cos  57.29578°,  or 
approximately  cos  57.3°,  we  have 

, , 1111 

cos  1 = 1 1 — 1 ••• 

2 ! 4 ! 6 ! 8 ! 

= 1 - 0.5  -f-  0.04167  - 0.00139  + 0.00002  - • • • 

= 0.5403  = cos  57°  18'. 


163.  Euler’s  Formula.  An  important  formula  discovered  in  the 
eighteenth  century  by  the  Swiss  mathematician  Euler  will  now  be 
considered.  We  have,  as  in  § 162, 

/y>3  /yt? 

sinx  = x-:^  + ---  + ..; 


and 


/y»2  /y»4 

. *K/  *Ay 

cosa;  = l--  + ---  + .... 


ix*  ix} 


By  multiplying  by  i in  the  formula  for  sin  x,  we  have 
i sin  x — ix 

Adding, 

/y»2  'i "3 

( • . , tK/  l/vU  vU  fjUL/ 

COS  X + ^ sm  X = 1 + ^x  — — — — + — -f-  — — . . .. 

By  substituting  ix  for  x in  the  formula  for  we  see  that 

i^x?  iV  tV  i®x® 

0”  = l + “+  2T  + 3T+  4! 

/y»2  ^ 'j  ■y*^ 

j , iL  tfiAy  tAy  t/tAy 

In  other  words, 

e“  = cos  x + i sin  x. 


182 


PLANE  TEIGONOMETRY 


164.  Deductions  from  Euler’s  Formula.  Euler’s  formula  is  one  of 
the  most  important  formulas  in  all  mathematics.  From  it  several 
important  deductions  will  now  be  made. 

Since  e“  = cos  x + i sin  x,  in  which  x may  have  any  values,  we 
may  let  a:  = tt.  We  then  have 

e‘^  : cos  7T  + i sin  tt  = — 1 + 0, 
or  e'”  = — 1. 

In  this  formula  we  have  combined  four  of  the  most  interesting  numbers  of 
mathematics,  e(the  natural  base),  f (the  imaginary  unit,  V—  l),  7r(the  ratio  of 
the  circumference  to  the  diameter),  and  — 1 (the  negative  unit). 

Furthermore,  we  see  that  a real  number  (e)  may  be  afiected  by  an  imaginary 
exponent  (itr)  and  yet  have  the  power  real  (—  1). 

Taking  the  square  root  of  each  side  of  the  equation  e’"'  = — 1, 
we  have  

= V—  1 = i. 


Taking  the  logarithm  of  each  side  of  the  equation  e'”’  = — 1, 
ITT  = log  (-1). 


Hence  we  see  that  — 1 has  a logarithm,  but  that  it  is  an  imaginary  number 
and  is,  therefore,  not  suitable  for  purposes  of  calculation. 


Since  cos  <f>  + i sin  (f>  = cos  (2  kir  + (f>)+  i sin  (2  + <f>),  we  see 

that  e**”’,  which  is  equal  to  cos  <f>  + i sin  <f>,  may  be  written  + 
or  we  may  write 

g<l>i  _ g(2t7r  + (i)i  _ ^ i sin  (j)  = cos  (2  fcTT  + <^)  + i sin  (2  Jctt  + <f). 

Hence  (2  kTr  + <^)  i = log  [cos  (2  kir  + <!>)+  i sin  (2  kTr  + <^)]. 

If  ^ = 0,  2 kiri  - log  1. 

If  fc  = 0,  this  reduces  to  0 = log  1. 

If  fc  = I we  have  2 rri  = log  1 ; if  k = 2,  we  have  4-rri  = log  1,  and  so  on.  In 
other  words,  log  I is  multiple-valued,  but  only  one  of  these  values  is  real. 

If  <^  = 7T,  (2  kir  + 7r)i  = (2  k + l)7ri  = log  (—  1). 

Hence  the  logarithms  of  negative  numbers  are  always  imaginary ; for  if  A:  = 0 
we  have  ttI  = log  (—  1) ; if  A:  = 1 we  have  Sm  — log  (—  1) ; and  so  on. 

If  we  wish  to  consider  the  logarithm  of  some  number  A,  we  have 
= N (cos  2 A'tt  -b  i sin  2 Att). 

Hence  log  A -f  2 kiri  = log  N -t-  log  (cos  2 Att  -f-  i sin  2 krr) 

= log  A -f-  log  1 = log  N. 

That  is,  log  A = log  A -|-  2 kiri.  Hence  the  logarithm  of  a number  is  the 
logarithm  given  by  the  tables,  -J-  2 km.  If  A = 0 we  have  the  usual  logarithm, 
but  for  other  values  of  k we  have  imaginaries. 


APPLICATIONS  TO  ALGEBEA 


188 


Exercise  83.  Properties  of  Logarithms 


Prove  the  following  properties  of  logarithms  as  given  in  § 159, 
using  h as  the  base : 

1.  Properties  1 and  2.  3.  Property  4.  5.  Property  6. 

2.  Property  3.  4.  Property  5.  6.  Property  7. 


Find  the  value  of  each  of  the  following  : 
7.  5!  8.  7! 

Simplify  the  following : 


12. 


10! 
3!  ■ 


13. 


10! 
8!  ' 


9. 

6! 

10.  8! 

11. 

10! 

7! 

15! 

20! 

14. 



15.  — — • 

16. 

— 

5! 

14! 

17! 

/ 1 1 

17.  Pind  to  five  decimal  places  the  value  of  (1+1+  ^ ^ + ' ' ' ) • 

/II 

18.  Find  to  five  decimal  places  the  value  ^ ^ ^ + ' ' ’ ) • 

By  the  use  of  the  series  for  cos  x find  the  following : 

20.  cos  21.  cos  2.  22.  cos  0. 


By  the  use  of  the  series  for  sin  x find  the  following : 

23.  sinl.  24.  sin  25.  sin  2.  26.  sinO. 

By  the  use  of  the  series  for  tan  x find  the  following : 

27.  tan  0.  28.  tan  1.  29.  tan  J.  30.  tan  2. 

Prove  the  following  statements: 

7T  

31.  e'^”^  - 1.  32.  = i\  33.  s’"  = 34.  d = V—  1. 

Given  log = 0.6931,  find  two  logarithms  (to  the  base  e)  of: 

36.  2.  36.  4.  37.  V§.  38.  — 2. 


Given  log ^5  = 1.609,  find  three  logarithms  (to  the  base  e)  of: 

39.  5.  40.  25.  41.  125.  42.  - 5. 


Given  logfiO  = 2.302585,  find  two  logarithms  (to  the  base  e)  of: 
43.  100.  44.  — 10.  45.  1000.  46.  VTO. 

47.  From  the  series  of  § 162  show  that  sin(—  <^)  = — sin  <f>- 

48.  Prove  that  the  ratio  of  the  circumference  of  a circle  to  the 
diameter  equals  — 2 log  (i*)  = — 2 i log  i. 


184 


PLAISTE  TEIGONOMETEY 


Exercise  84.  Review  Problems 

1.  The  angle  of  elevation  of  the  top  of  a vertical  cliff  at  a point 
575  ft.  from  the  foot  is  32°  15'.  Find  the  height  of  the  cliff. 

2.  An  aeroplane  is  above  a straight  road  on  which  are  two  observers 
1640  ft.  apart.  At  a given  signal  the  observers  take  the  angles  of  ele- 
vation of  the  aeroplane,  finding  them  to  be  58°  and  63°  respectively. 
Find  the  height  of  the  aeroplane  and  its  distance  from  each  observer. 

3.  Prove  that  (Vcscx  + cotx  — Vcsca:  — cotx)^  = 2 (esc  a;  — 1). 

4.  Given  sina:  = 2 rn/(m?  + 1)  and  sin  y = 2 n/(r^  1),  find  the 

value  of  tan(£c  -F  y). 

5.  Find  the  least  value  of  cos'^a:  + sec^a;. 

6.  Prove  that  1 — sin^a:/sin^y  = cos^a;(l  — tan^a:/tan^y). 

7.  Prove  this  formula,  due  to  Euler ; tan~^^  -f  tan~^^  — 

8.  Prove  that  cot  — cot x = esc x. 

9.  Prove  that  (sin  x + i cos  a;)"  = cos  n(^7r  — x)  + i sin  nQvr  — x). 

10.  Show  that  log  i J 'rri  and  that  log  (—  i)  = — |.  tti. 

11.  Through  the  excenters  of  a triangle  ABC  lines  are  drawn 
parallel  to  the  three  sides,  thus  forming  another  triangle  A'B'C’. 
Prove  that  the  perimeter  of  AA'B'C'  is  4 r cot  cot  cot -^C, 
where  r is  the  radius  of  the  circumcircle. 

12.  Given  two  sides  and  the  included  angle  of  a triangle,  find 
the  perpendicular  drawn  to  the  third  side  from  the  opposite  vertex. 

13.  To  find  the  height  of  a mountain  a north-and-south  base  line  is 
taken  1000  yd.  long.  From  one  end  of  this  line  the  summit  bears 
N.  80°  E.,  and  has  an  angle  of  elevation  of  13°  14' ; from  the  other 
end  it  bears  N.  43°  30'  E.  Find  the  height  of  the  mountain. 

14.  The  angle  of  elevation  of  a wireless  telegraph  tower  is  observed 
from  a point  on  the  horizontal  plain  on  which  it  stands.  At  a point  a 
feet  nearer,  the  angle  of  elevation  is  the  complement  of  the  former. 
At  a point  b feet  nearer  still,  the  angle  of  elevation  is  double  the  first. 
Show  that  the  height  of  the  tower  is  [(a  + by  — ^ 

Prove  the  following  formulas : 

15.  2cos*a:  = cos  2a;-t-l.  17.  8 cos^jc  = cos  4x -f  4cos  2 a;  + 3. 

16.  2 sin^a;  = — cos  2 a;  + 1.  18.  4 cos® a:  = cos  3a;  + 3 cos  a:. 

19.  4 sin®x  = — sin  3 a;  + 3 sin  x. 

20.  8 sin^a;  = cos  4 a;  — 4 cos  2 x + 3. 


FOKMULAS 


185 


THE  MOST  IMPOETANT  FOEMULAS  OF  PLANE 
TEIGONOMETEY 

Eight  Triangles  (§§  15-21) 

1.  y = r sin  <p.  A.  x = y cot 

2.  x = r cos  5.  r — X sec  <ji. 

3.  y = X tan  <j>.  6.  r = y esc  (f>. 

Eelations  of  Functions  (§§  13,  14,  89) 


7.  sin  ^ = 

8.  cos  fj)  = 

9.  tan  <l>  = 


CSC  4> 

1 

sec  <j) 
1 

cot  ^ 


12.  cot  (f>  — 

13.  sec  <f> 

1 -1.  CSC  4>  = 


tan  (ji 
1 
COS 

1 

sin  (j> 


17.  sin  <l>  = 

18.  tan  ^ = 


cos  <}> 

COt<fi 

sin  (f) 
cos  <l> 


, „ , , cos  d> 

19.  cot  <f>  — — ^ 


sin  <f> 

10.  sin^csct^=l.  15.  tan  c/>  cot  ^ = 1.  20.  1 +tan^<^  = sec^<^. 

11.  cos^sec<^=l.  16.  sin^(^  + cos^^=l.  21.  1 + cot^<^  = csc^^. 

Functions  ofx  ±y  (§§  90-100) 

22.  sin  (x  + y)=  sin  x cos  y + cos  x sin  y. 

23.  sin  (x  — y)=  sin  x cos  y — cos  x sin  y. 

24.  cos  (x  -\-y)=  cos  x cos  y — sin  x sin  y. 

25.  cos  {x  — y)=  cos  x cos  y + sin  x sin  y. 

, tan  a;  4- tan?/  x coticcot?/  — 1 

^ ± — tan  X tan  y \ y _j_  ^ 

^ . . tan cc  — tan?/  cot x cot  ?/ 4- 1 

27.  tan(a;  — w)=- 29.  cot(a:  — ?/)  = — : 

^ ' l + tana:tan^  ' cot?/  — cota; 

Functions  of  Twice  an  Angle  (§  101) 

30.  sin  2 <l>  = 2 sin  (f>  cos  (j>.  32.  cos  2 (f>  = cos^<fi  — sin^^. 


31.  tan  2<j>  = 


2 tan 


33.  cot  2<j>  = 


cot^  <^  — 1 


1 — tan^  <j)  * 2 cot  4> 

Functions  of  Half  an  Angle  (§  102) 

36.  tan  ^ = ± 


cos  4> 


34.  sin^^  =±^ 

h + cos  d> 
36.  cosi<f>=±^- 


1 — cos 


37.  cot^^ 


1 + cos  ^ 


cos  <f> 


cos  <f> 


186 


PLANE  TRIGONOMETRY 


Functions  involving  Half  Angles  (§  101) 


c.  ■ X X 

38.  sin  X = 2 sm  - cos  - • 

^ Jj 


40.  COS  X = cos^-  — 

Jj 


■ 2^ 
sin^-- 


39.  tan  X — 


2 tan^ 


41.  cot  X = 


2cot^ 


Sums  and  Differences  of  Functions  (§  103) 

42.  sinal  + sin  A = 2 sin  ^(^4  + A)  cos  — B). 

43.  sin^I  — sin 5 = 2 cos  ^(A  + A)  sin  \(A  — B). 

44.  cosal  + cos  A = 2 cos  ^(A  + 'A)cos  ^(^1  — A). 

45.  cosal  — cos  A — — 2 sin  + A)  sin  ^(^4  — A), 

sin  A + sin  A _ tan  ^ (^4  + A) 


46. 


sina4  — sin  A tan  ^ (^4  — A) 


Laws  of  Sines,  Cosines,  and  Tangents  (§§  105,  111,  112) 

a _ sinyl 
b sin  A 
a h c 


47.  Law  of  sines. 


^ 48.  Law  of  cosines, 
^ 49.  Law  of  tangents. 


a + & + c 
50.  ^ — = s. 


sin^l  sin  A sinC 

= b^  — 2 be  cos  A . 

a — b tan  1 (.4  — A)  ^ 

7 = ■; , 7 » if  a > 0 ; 

a b tan  (^.4  -|-  A) 

b — a tanl(A— -4)  .. 

= TV 7 ) if  G < 0. 

b + a tan  -^A  + .4  ) 


Formulas  in  Terms  of  Sides  (§§  115,  116) 

53 


-a){s-b)(s-c)  ^ 

s 


51. 

62.  cos  ^A  = 


j(s  — b)(s  — c) 

54.  tan  LG  = ^ • 

^ > s(s  — a) 


js(s  — a) 
be 


55.  tan  ^A  -- 


Areas  of  Triangles  (§  118) 

56.  Area  of  triangle  *4  AC  = ^ ac  sin  A = ^ ?■(«  + 5 + c)  = ra  = 

, abc  sin  A sin  C 

Vs(s  _ a) (s  - 5) (s  - c)  = — = 


4 A 2 sin  (A  + C) 


INDEX 


PAGE 

Abscissa 78 

Addition  formulas  ....  97,  101 

Algebra,  applications  to  . . . . 173 

Ambiguous  case 112 

Angle,  functions  of  an  ....  3,  4 

of  depression 18 

of  elevation 18 

negative 77,  92 

positive 77 

Angles,  difference  of 100 

differing  by  90° 92 

greater  than  360° 87 

having  the  same  functions  154, 155 

how  measured 2 

sum  of 97 

Autilogarithm 48 

Areas  ......  66,  128,  141,  142 


Base '40 

Briggs 39 

Changes  in  the  functions  ...  25 

Characteristic 43 

negative 44,  61 

Circle 144 

Circular  measure 151 

Cologarithm 54 

Compass  146 

Complementary  angles  ....  7 

Conversion  table 30 

Coordinates 78 

Cosecant 4,  22 

Cosine 4,  16,  116,  180 

Cosines,  law  of 116 

Cotangent 4,  20 

Course 145 

Coversed  sine 171 


. . . . 30 

. ...  174 
. ...  145 

187 


PAGE 

Depression,  angle  of 18 

Difference  of  two  angles  ....  100 

of  two  functions 105 

Division  by  logarithms  ...  42,  52 

Elevation,  angle  of 18 

Eliminant 171 

Equation  ....  163,  166,  169,  173 

Euler 181 

Euler’s  Formula 181 

Expansion  in  series 180 

Exponential  equation 58 

series 179 

Formulas,  important 185 

Fractional  exponent 57 

Functions  as  lines 23 

changes  in  the 25 

graphs  of 158 

inverse 156. 

line  values  of 85 

logarithms  of 60 

of  a negative  angle  ....  92 

of  an  angle 3,  10 

of  any  angle 82 

of  half  an  angle  . . . 104, 123 

of  small  angles 153 

of  the  difference  of  two  angles  100 
of  the  sum  of  two  angles  . . 97 

of  30°,  4.5°,  60° 8 

of  twice  an  angle  . . . . . 103 

reciprocal 12 

relations  of 12,  13 

variations  in 86 

Graphs  of  functions 158 

Half  angles 104,  123 


Decimal  table  . . . 
De  Moivre’s  Theorem 
Departure  .... 


Identity  . . 
Interpolation 


. . 163 
31,  32,  48 


188 


INDEX 


PAGE 

Inverse  functions 156 

Isosceles  triangle 70 

Latitude 145 

Laws  of  the  characteristic  ...  44 

of  cosines 116 

of  sines 108 

of  tangents 118 

Logarithm 40 

Logarithms 39 

of  functions 60 

properties  of 178 

systems  of 178 

use  of  tables  of  ....  46,  61 

Mantissa 43 

Middle  latitude  sailing  ....  149 

Multiplication  by  logarithms  . 42,  50 

Napier  39 

Negative  angle 77,  92 

characteristic 44,  51 

lines 77 

Oblique  angles 77 

triangle 107 

Ordinate 78 

Origin 78 

Parallel  sailing 148 

Plane  sailing  . 145 

trigonometry 1 

Polygon,  regular 72 

Positive  angle 77 

Power,  logarithm  of  ....  43,  56 

Practical  use  of  the  cosecant  . . 22 

of  the  cosine 16 

of  the  cotangent 20 

of  the  secant 21 

of  the  sine 14 

of  the  tangent 18 


PAGE 

Quadrant 78 

Radian 151 

Reciprocal  functions 12 

Reduction  of  functions  to  first 

quadrant 90 

Regular  polygon 72 

Relations  of  the  functions  . 12,  13,  94 

Right  triangle 34,  63, 133 

Root,  logarithm  of 43,  57 

Roots  of  numbers 177 

of  unity 175 

Secant 4,  21 

Series,  exponential 179 

Sexagesimal  table 28 

Signs  of  functions 86 

Simultaneous  equations  ....  169 

Sine 4,  14,  108,  180 

Sines,  law  of 108 

Sum  of  two  angles 97 

of  two  functions 105 

Surveyor’s  measures 142 

Symbols 3,  4,  40,  171 

Tables  explained  10,  28,  30,  46,  48,  61 

Tangent 4,  18 

Tangents,  law  of 118 

Traverse  sailing 150 

Trigonometric  equation  ....  163 

identity 163 

Trigonometry,  nature  of  ...  . 1 

plane 1 

Unity,  roots  of 175 

Variations  in  the  functions  . . 86 

Versed  sine  171 


AJN^SWERS 


ANSWEES 


PLANE  TRIGONOMETRY 


Exercise  1.  Page  5 

1.  cos5=-;  tan5=-;  cotB=~\  seoB^-;  csc-B=-- 
c a b a b 

4.  cotJ..  5.  sec^.  6.  cscA. 

I ; cos-4  = i ; tan  A = | ; cot^  = | ; sec  f ; csc^  = 

■j®g- ; cos-4  = i| ; tan-4  = ; cot-4  = JJ. ; sec-4  = 1 1 ; csc^  = 

; cos-4  = jj',  tan-4  = Jg  ; cot-4  = Jg®- ; sec-4  = ; csc-4  = 

; ccs-4  = I ° ; tan  A = cot  A = ; sec  -4  = ; esc  -4  = 

4;  cos-4  = M;  tan-4  = j cot-4  = |g;  sec-4  = ; csc-4  = 

; cos-4  l|a  ; tanJ.  = 119  ; cot^  = ifg  ; sec4  = {W> 


15. 

16. 

ir. 

19. 

20. 

21. 

22. 

23. 

24. 


tan-1, 
sin -4  = 
sin -4  = 
sin -4  = 
sin -4  = 
sin-1  = 
sin-1 
(;■'  -1  = 
-f.  &2 

sin-1  = 
sec -4  : 
sin-4  : 
sec -4  : 
sin -4  : 
sec -4 : 
sin -4  = 

sec-1  = 

sin -4  = 
sin -4  = 
csc-4  = 
sin -4 : 
csc-1 : 
sinl? 
CSC  B 
sinB 
csc£ 
sinB 
cscB 


3 „ 

■gs 
1 1 ^ 

T6  9 
1B9 
119’ 

= C2. 

2n 
+ l’ 
+ 1 
n^  — 1 
2n 

Tl^  + 1 ’ 
M-  + 1 
— 1 
2mn 


; cos-1  = 
csc-1  = 
cos-1  = 
csc-1  -- 


vP- 

— 

1 

vP 

+ 

1 

vP 

+ 

1 

2n 

vP 

- 

1 

vP 

+ 

1 

4- 

1 

tan-1 


2n 
— 1 


; cot -4 


2n 


tan -4  = 


2n 

n'^  — 1 


; cot-1  = 


— 1 
2n 


— 1 
2 n 


w?  + V? 

w?  + 
m?  —V?’ 
2mn 
w?  + ’ 

+ V? 


- ; cos-1  = 
csc-4  = 
cos-1  = 
; csc-1  = 


m2 

- 

m2 

mP' 

+ 

rP 

m2 

+ 

«2 

2 

mn 

m2 

— 

n2 

m2 

+ 

n2 

mP 

+ 

n2 

: tan-4 


: tan -4  = 


2mn 


2mn 

n ,2  — 7)2  ' 


cot  A = 


n%  — 


2 mn 


cot -4  = 


2 mn 


^2  _ jj2  2 mn 

1 V2  = cos-4  ; tan^  = 1 = cot-4  ; sec-4  = Vi  = csc-4. 

§■  V5;  cos-4  = 4 Vs  ; tan-4  = 2 ; cot-4  = ^ ; sec-4  = Vs  ; 


iVs. 


; cos-4  = ^ Vs ; tan-4  = |V5;  cot-4  = -^VS;  sec -4  = 3- VS; 
cos  B = yW ; tan  B = 1/^1 ; cot  B = ; sec  B = lj\5. ; 

cosB=  jfl;  tanB  = ^^^s-,  cotB  = J^e, 


• 3 

- 3 

- J- 

-143. 

■ T4  5 > 

- 1 4 S 

■ T4^ 


secB=  III; 


= cosB=||-4;  tanB=j2^\;  cotB  = 2^^--,  seoB  = ||f; 

:-W- 


1 


2 


PLANE  TRIGONOMETRY 


26.  sinR  = 
seoR  = 

26.  sin  A 

cos  A = 

tan  A = 

27.  sin  A 

Co:-,  si  = 


cosR  = ^; 
P+Q  P+q 


tanR  = 


2 

p-q  ’ 


cotB  = 


p + q. 
p-q' 

Vp2  + ( 

+ 9 

P + g 
Vp2  + 
\/2pq 

-v^2 ^ p 

p + 1 

1 

Vp  + 1 


cscR  = 


+ g 

2pg 


• = cos  jB  ; cot  A = 


Vp^  + 


: = tan  B ; 


= sin  B ; 


.sec A = ^ = cscB : 


= cotB;  CSC  A = 


Vp2  + 


: = secR. 


= cosR : 


= sinR ; 


Vp 

cot  A = — — = tan  R ; 


;A  - Vp  + 1 = cscR; 


tan  A = 

Vp 

= cotR ; 

CSC  A = 

+ p „ 

— = — — sec  R. 

28. 

12.3. 

37. 

2.5;  1.5. 

47. 

a = 4.501  ; h = 5.362. 

29. 

1.64. 

38. 

1.5mi.  ; 2 mi. 

48. 

a = 6.8801 ; 6 = 8.1962. 

30. 

9. 

40. 

a = 0.342  ; 

6 = 0.94. 

49. 

a = 160.75;  6 = 191.-5. 

31. 

6800. 

41. 

a = 1.368; 

h = 3.76. 

50. 

a = 1.88  ; 6 = 0.684. 

32. 

4000. 

42. 

a = 1.197  ; 

h = 3.29. 

51. 

c = 2.128;  6 = 0.728. 

33. 

227.84. 

43. 

a = 1.6416 

; 6 = 4.512. 

52. 

c = 5.848  ; a = 5.494. 

34. 

3V13;  9. 

44. 

a = 2.565  ; 

6 = 7.05. 

53. 

c =26.6;  6=  9.1. 

35. 

45. 

a = 0.643  ; 

h = 0.766. 

54. 

a = 412.05;  c =438.6. 

36. 

5;  3. 

46. 

a = 1.929; 

h = 2.298. 

55. 

142.926  yd. 

56. 

11  ; 24  ft. 

Exercise  2.  Page  7 


1. 

cos  60°. 

5.  COS  40°. 

9.  cos  30°. 

13. 

cos  14°  30'. 

17. 

cos  2-5°.  21. 

tan  29°, 

2. 

sin  70°. 

6.  cot  30°. 

10.  sin  30°. 

14. 

cot  7°  15'. 

18. 

cot  10°.  22. 

sec  12°. 

3. 

cot  50°. 

7.  CSC  15°. 

11.  cot  45°. 

15. 

CSC  21°  45'. 

19. 

CSC  13°.  23. 

cos  1°. 

4. 

CSC  65°. 

8.  sec  5°. 

12.  CSC  45°. 

16. 

sin  1°  50'. 

20. 

sin  38°.  24. 

sin  4°. 

25.  CSC  2°. 

27.  sin7-t° 

29.  45°. 

31. 

30°. 

26.  cos  12^°. 

28.  cot  1.4°. 

CO 

p 

32. 

30°. 

Exercise  3. 

Page  9 

1. 

0.5. 

5. 

1.1547.  9.  1.7320. 

13.  V2. 

17. 

21. 

2. 

0.8660. 

6. 

2.  10.  0.5773. 

14.  We. 

18. 

;^V2.  22. 3. 

3. 

0.5773. 

7. 

0.8660.  11.  2. 

15.  V3. 

19. 

IV3.  23. 

4. 

1.7320. 

8. 

0.5.  12.  1.1547. 

16.  IV3. 

20. 

A 3.  24. 

25. 

cos  27°  42' 

'20".  27.  CSC  2°  27' 9". 

29.  COS  14.2°. 

31.  cot  21.18°. 

26. 

cot  14°  31' 25'' 

'.  28.  sin  1°  59'  33". 

30.  sin 

7.25°. 

32.  CSC  4.05°. 

33. 

90°. 

37. 

90°  40.  22°  30'. 

43.  jVe. 

44.  V2. 

47. 

2a'3.  51.  1. 

2.  68.  iVs. 

34. 

60°. 

n+1  41.18°. 

48. 

35. 

36. 

22°  30'. 
18°. 

38. 

39. 

90°.  42  90° 

60°.-^l  ■ ■a  + 1 

45.  Ve. 

46.  fVi. 

49. 

50. 

iVs. 

^V3. 

ANSWEJIS 


3 


Exercise  4.  Page  10 


1. 

0.0872. 

7.  0.3584. 

13. 

0.9136.  19. 

5.1446. 

25.  1.0000. 

31. 

1.4396. 

2. 

0.2419. 

8.  0.5000. 

14. 

0.9135.  20. 

5.1446. 

26.  1.0000. 

32. 

1.4396. 

3. 

0.3684. 

9.  0.9945. 

15. 

0.8192.  21. 

0.3839. 

27.  1.0353. 

33. 

0.0038. 

4. 

0.5000. 

10.  0.9945. 

16. 

0.8192.  22. 

0.3839. 

28.  1.0353. 

34. 

0.0054. 

5. 

0.0872. 

11.  0.9703. 

17. 

11.4301.  23. 

1.0000. 

29.  4.8097. 

35. 

2 sec  10°. 

6. 

0.2419. 

12.  0.9703. 

18. 

11.4301.  24. 

1.0000. 

30.  4.8097. 

36. 

2 CSC  10°. 

37.  2 cos  15°. 

38.  3 sin  20°  > sin  (3  x 20°)  and  > sin  (2  x 20°). 

39.  3 tan  10°  < tan (3  x 10°)  and  > tan (2  x 10°). 

40.  3 cos  10°  > cos  (3  x 10°)  and  > cos (2  x 10°). 

41.  No. 

42.  The  sin,  tan,  sec  increase  and  the  cos,  cot,  esc  decrease. 

Exercise  5.  Page  12 


12. 

37.6. 

13.  1. 

14.  100. 

15.  60. 

16.  12.86. 

17.  22.64. 

Exercise  6. 

Page  15 

1. 

1.736. 

4.  57.45, 

7.  39° 

10. 

54 

ft. 

13.  449.9  ft. 

2. 

3.882. 

5.  12°. 

8.  43° 

11. 

4.326  ft. 

3. 

41.01. 

6.  20°. 

9.  30° 

12. 

479.9  ft. 

Exercise  7. 

Page  16 

1. 

10.83. 

8. 

5.935. 

15.  63°.  22. 

411.4  ft. 

29.  6 in. 

2. 

13.46. 

9. 

4.884. 

16.  70°.  23. 

383  ft. 

30.  28.19ft.;  21.21  ft.; 

3. 

25.58. 

10. 

7.311. 

17.  54°.  24. 

43°. 

12.68 ft.;  30 ft.;  0ft 

4. 

31.86. 

11. 

10°. 

18.  60°.  25. 

7.794  in. 

31.  60°; 

0°. 

5. 

55.73. 

12. 

17°. 

19.  70°.  26. 

166.272  sq.  in. 

32.  2.5°; 

65°. 

6. 

1.873. 

13. 

26°. 

20.  84°.  27. 

5.657. 

33.  30°  and  60°; 

7. 

5.972. 

14. 

60°. 

21.  60°.  28. 

27.71  ft. 

31°  and  59°. 

34.  749.  £ 

Ift. 

Exercise  8 

. Page  19 

1. 

12.02. 

6. 

5.928. 

11.  45°. 

16.  64°. 

20. 

159.7  ft. 

2. 

11.04. 

7, 

, 14.78. 

12.  8°. 

17.  148  ft. 

8 in.  21. 

45°;  90°;  45°. 

3. 

28.84. 

8. 

44.01. 

13.  9°. 

18.  29°. 

22. 

15.76  ft. 

4. 

45.04. 

9. 

107.1. 

14.  19°. 

19.  2.517 mi; 

23. 

6.14  ft. 

5. 

98. 

10. 

453.8. 

15.  22°. 

3.916  mi. 

24. 

1.03  in. 

Exercise  9. 

Page  20 

1. 

26.11. 

4.  85.81. 

7.  26.60.  10. 

26' 

) 

13.  113  ft. 

2. 

12.36. 

5.  544.0. 

8.  68.80.  11. 

28. 

87  ft. 

14.  123.6  ft 

3. 

162.6. 

6.  26.84. 

9.  45°. 

12. 

428.4  ft. 

Exercise  10.  Page  21 

1. 

40.40. 

4.  33.63. 

7.  41°. 

10. 

57. 

.74  ft. 

13.  26.11ft 

2. 

61.77. 

5.  65.50. 

8.  60°. 

11. 

1369  ft. 

3. 

101.2. 

6.  339.4. 

9.  22.66  ft.  12. 

91, 

,64  ft. 

4 


PLANE  TRIGONOMETRY 


Exercise  11.  Page  22 


1.  49,60.  3.  80.62.  5.  81.19. 

2.  54.87.  4.  64.60.  6.  152.8. 

13.  19.82  mi.  14.  267.0  ft. 


7.  64°.  9.  65°.  11.  1113  ft. 

8.  28°.  10.  45°.  12.  13.69  mi 

15.  57.51ft.  16.  17.23  in. 


Exercise  12.  Page  23 

3.  tana;.  4.  secx.  5.  secx.  6.  cscx.  7.  cotx.  8.  cscx.  16.18°.  35.  rsinx. 
36.  a = cm;  b = c Vl  — m^.  37.  a = bm;  c = b + 1. 


Exercise  13.  Page  26 


2.  0. 

8.  No. 

13.  2.3109. 

19.  37°. 

25.  19°. 

31. 

16^ 

3.  1. 

9.  45°. 

14.  0.5373. 

20.  46°. 

26.  48°. 

32. 

31^. 

4.  00. 

10.  0.6462; 

15.  6°. 

21.  6°. 

27.  34°. 

33. 

1 

5* 

5.  0. 

0.7631. 

16.  24°. 

22.  13°. 

28.  40°. 

6.  The  tangent. 

11.  0.3680. 

17.  44°. 

23.  22°. 

29.  54°. 

7.  No. 

12.  2.7173. 

18.  26°. 

24.  14°. 

30.  30°. 

Exercise  14.  Page  29 


1. 

0.7647. 

7. 

0.7428. 

13.  0.8708. 

19. 

53.47. 

25. 

69.38. 

31. 

19.70  ft.; 

2. 

0.9004. 

8. 

0.6563. 

14.  0.8708. 

20. 

20.90. 

26. 

49.83. 

22.62  ft. 

3. 

0.7545. 

9. 

0.6693. 

15.  1.1483. 

21. 

25.27. 

27. 

94.35. 

32. 

19.72  ft.; 

4. 

0.9015. 

10. 

0.6567. 

16.  17.73. 

22. 

48.29. 

28. 

74.93. 

22.61  ft. 

5. 

0.7538. 

11. 

0.6700. 

17.  32.16. 

23. 

66.48. 

29. 

88.35, 

33. 

120.5  ft. 

6. 

0.7546. 

12. 

0.6700. 

18.  46.01. 

24. 

64.84. 

30. 

47°  56'. 

34. 

71.77  ft. 

Exercise  15.  Page  30 


1.  0.0087. 

6.  0.0715. 

11.  0.9972. 

16.  1.0000. 

21.  12.66  in.  ; 

2.  0.0070. 

7.  0.9972. 

12.  0.9974. 

17.  0.0715. 

0.9970  in. 

3.  0.0698. 

8.  0.0769. 

13.  0.0767. 

18.  143.2. 

22.  390  ft. 

4.  0.9973. 

9.  12.71. 

14.  13.95. 

19.  0.0052. 

23.  0.7477  in. 

5.  0.0787. 

10.  13.62. 

15.  0.0769, 

20.  0.0734. 

9.530  in. 

Exercise  16.  Page  33 


1. 

0.4567. 

14.  12.1524. 

24.  70°  45' 30"; 

35.  10.7389. 

48. 

44°  38' 30" 

2. 

0.6725. 

15.  15.3140. 

0.3490. 

36.  0.9808. 

49. 

69°  15'. 

3. 

0.8338. 

16.  10.4652. 

25.  79°  30' 15"; 

37.  4.5787. 

50. 

78°  8'  30". 

4. 

0.9099. 

17.  8.7149. 

0.1852. 

38.  4.1525. 

51. 

78°  8' 15". 

5. 

0.8065. 

18.  7.2246. 

26.  0.4305. 

39.  3.6108. 

52. 

14°  45'. 

6. 

0.7289. 

19.  6.6585. 

27.  0.4313. 

40.  3.3502. 

53. 

0.7658. 

7. 

0.4335. 

20.  6.0826. 

28.  0.5410. 

41.  31°  30'. 

54. 

0.6438. 

8. 

0.5438. 

21.  39°  43' 30"; 

; 29.  0.6646. 

42.  35°  15'. 

55. 

0.5639. 

9. 

0.6418. 

0.7691. 

30.  0.9045. 

43.  41°  18' 30". 

56. 

33°  10'  15". 

10. 

0.9209. 

22.  50°  16' SO"; 

31.  0.1990. 

44.  44°  36' 30" 

1.5298. 

11. 

1.2882. 

0.6391. 

32.  4.9550. 

45.  38°  15'. 

57. 

31°  8'  30"; 

12. 

2.5018. 

23.  71°  29' 40"; 

; 33.  0.1490. 

46.  39°  30'. 

0.6042. 

13. 

3.1266. 

0.9483. 

34.  7.8279. 

47.  17°  46'. 

ANSWERS 


5 


Exercise  17.  Page  37 


1.  A = 36°62',  R=53°8',  c = 5. 

2.  A = 32° 35',  R=  67° 25',  6 = 10.95. 

3.5  = 77°  43',  b = 24.34,  c = 24.93. 

4.  A = 46°  42',  6 = 9.801,  c = 14.29. 

5.  5 = 52°  18',  a = 15.90,  6 = 20.57. 

6.  A = 65° 48',  a = 127.7,  6 = 57.39. 

7.  A = 34°  18',  5=  5-5° 42',  a = 12.96. 

15.  5 = 51°  31',  a 

16.  A = 22°  37',  B 

17.  A = 53°  8',  B 

18.  A = 22°  37',  B 


8.  A = 43°  33',  5=46°27',a  = 93.14. 

9.  5 = 67°  46',  a = 26. 73,  c = 50.12. 

10.  A = 43° 49',  a = 191.9,  c = 277.2. 

11.  A = 68° 43',  5=21°17',  c = 102.0. 

12.  A = 3°  20',  B = 86°  40',  6 = 102.8. 

13.  A = 84°  52',  6 = 0.2802,  c = 3.133. 

14.  A = 70°  48',  5=19°  12',  6 = 5.916. 
35.47,  6 =44.62. 

67°  23',  a = 5,  c = 13. 

36°  52',  a = 40,  c = 50. 

67°  23',  a = 12.5,  c = 32.5. 


19.  5 = 54° 49' 30",  6 = 3.547,  c=  4.340.  21.  A = 60° 41' 30",  6=3.593,  c = 7.339. 

20.  5 = 47°47'30",  6 = 6.284,  c=8.485.  22.  A = 63°  39' 30",  6=5.812,  c=9.808. 

23.  5 = 60°  17'  30",  a = 3.370,  6 = 5.906. 

24.  5 = 55°  39'  30",  a = 203.08,  6 = 297.25. 

25.  5 = 48°  49' 20",  a = 218.68,  c = 332.14. 

26.  5 = 64.5°,  6 = 100.6,  c = 111.5. 


27.  5 = 65.5°,  a = 10.37,  6 = 22.76. 

28.  5 = 57.45°,  a = 21.52,  6 = 33.72. 

29.  5 = 34.49°,  a = 65.94,  6 = 45.30. 


30.  5 = 26.54°,  a = 67.10,  6 = 33.51. 

31.  A = 39.41°,  6 = 54.77,  c = 70.88. 

32.  B = 21.75°,  a = 225.6,  c = 242.8. 


33.  29.20  in. 

34.  23.73  in. 

35.  42.25  in. 

36.  64.26  in. 


37.  43.30  in. 

38.  60.05  in. 

39.  66°  18'  36",  33°  41'  24". 

40.  A = 41°  24'  30",  B = 48°  35'  30". 


41.  13.26  ft. 

42.  16.82  in.;  18.50  in. 

43.  12.42  ft. 

44.  66.89  in. 


45.  9°  35' 40". 


Exercise 

18. 

Page  41 

1. 

5. 

3.  4. 

5.  6. 

7. 

8.  9.  6. 

11. 

3. 

13.  3. 

15. 

4. 

17. 

3. 

19.  6. 

2. 

2. 

4.  4. 

6.  7. 

8. 

5.  10.  4. 

12. 

2. 

14.  3. 

16. 

2. 

18. 

6. 

20.  -1. 

21. 

— 

2;  -3; 

- 4. 

24. 

1;  2;  3 

; 6 

; 9; 

10: 

; — 

2;  -4; 

22. 

1 

and  2 ; 

2 

and  3 

; 3 and  4 ; 

-5;  - 

6;  - 

-7 ; 

- 8 

4 

and  5 ; 

5 

and  6 

; 8; 

and  9. 

25. 

1 ; 4;  6 

; 7; 

8 ; 

- 1 

) 

2;  -3; 

23. 

— 

2 and 

— 

1; 

3 and  — 2 ; 

-4;  - 

5 ; - 

- 6; 

- 7 

— 

4 and 

— 

3;  - 

1 and  0 ; 

26. 

0;  -4; 

C 

7 

; 8. 

— 

2 and 

1;  - 

3 and  — 2. 

27. 

1 

and  2. 

31. 

2 and  3. 

35.  3 and  4. 

39. 

5 and  6. 

28. 

1 

and  2. 

32. 

2 and  3. 

36.  3 and  4. 

40. 

6 and  7. 

29. 

1 

and  2. 

33. 

2 and  3. 

37.  3 and  4. 

41. 

6 and  7. 

30. 

1 

and  2. 

34. 

2 and  3. 

38.  3 and  4. 

42. 

7 and  8. 

Exercise 

19. 

Page  45 

1. 

1. 

6.  3. 

11.  - 

1. 

16. 

- 4. 

21. 

1.58681. 

2. 

1. 

7.  2. 

12.  - 

2. 

17.  ■ 

- 3. 

22. 

0.58681. 

3. 

2. 

8.  1. 

13.  - 

1. 

18.  • 

- 5. 

23. 

2.58681. 

4. 

0. 

9.  0. 

14.  - 

1. 

19. 

- 1. 

24. 

4.58681. 

5. 

3. 

10.  4. 

15.  - 

■ 3. 

20. 

- 2. 

25. 

5.58681. 

6 


PLANE  TRIGONOMETRY 


26. 

7.68681. 

32.  4.67724. 

38. 

T.40603. 

44.  1.39794, 

27. 

T.58681. 

33.  7.67724. 

39. 

3.40603. 

46.  2.39794. 

28. 

2.58681. 

34.  167724. 

40. 

4.40603. 

46.  4.39794. 

29. 

4.68681. 

35.  6.67724. 

41. 

7.40603. 

47.  7.39794. 

30. 

3.67724. 

36.  0.40603. 

42. 

0.39794. 

31. 

0.67724. 

37.  1.40603. 

43. 

1.39794. 

Exercise 

20. 

Page  47 

1. 

0.30103. 

14.  1.83556.  27. 

4.09157. 

40.  3.20732. 

53.  0.464.58. 

2. 

1.30103. 

15.  0.89905.  28. 

2.09157. 

41.  4.86198. 

54.  0.64167. 

3. 

2.30103. 

16.  2.92158.  29. 

2.37037. 

42.  0.48124. 

55.  1.08030. 

4. 

3.30103. 

17.  T.84510.  30. 

1.61624. 

43.  0.95424. 

56.  2.16224. 

5. 

3.32222. 

18.  1.87506.  31. 

1.75037. 

44.  0.90309. 

57.  0.79034. 

6. 

3.33244. 

19.  1.87852  . 32. 

1.61576. 

45.  4.22472. 

58.  1.14477. 

7. 

3.33365. 

20.  T.87892.  33. 

5.51409. 

46.  2.87595. 

59.  0.54254. 

8. 

0.33365. 

21.  2.40654.  34. 

2.56155. 

47.  5.32328. 

60.  0.99155. 

9. 

3.54220. 

22.  3.55630.  35. 

7.82948. 

48.  12.70040. 

61.  2.00072. 

10. 

3.64953. 

23.  4.95424.  36. 

17.72.562. 

49.  19.58460. 

62.  0.75343. 

11. 

3.74671. 

24.  2.25042.  37. 

9.19605. 

50.  0.15052. 

63.  1.19855. 

12. 

3.84663. 

25.  4.09132.  38. 

5.26893. 

51.  1.65052. 

13. 

3.72304. 

26.  4.09150.  39. 

2.51989. 

62.  1.17969. 

Exercise 

21. 

Page  49 

1. 

3. 

14.  7.6. 

27. 

6846.5. 

39. 

91.226. 

2. 

3000. 

15.  7,805,000,000. 

28. 

685.5.5. 

40. 

53,159,000. 

3. 

0.003. 

16.  79,950,000. 

29. 

77,553. 

41. 

0.000010745. 

4. 

304.5. 

17.  1.7102. 

30. 

785.65. 

42. 

5.72784; 

5. 

37,020. 

18.  27.005. 

31. 

7917.3. 

534,360. 

6. 

46. 

19.  370.15. 

32. 

8.5552. 

43. 

353,780. 

7. 

467.6. 

20.  0.38065. 

33. 

875.18. 

44. 

7.2388. 

8. 

0.000056. 

21.  0.0043142. 

34. 

2. 

45. 

107. 

9. 

6505. 

22.  43,144. 

35. 

3.45591  ; 

46. 

25,459. 

10. 

0.06796. 

23.  4.3646. 

3.45864. 

47. 

16,693,000. 

11. 

0.0006095.  24.  0.049074. 

36. 

2955. 

48. 

129.66. 

12. 

0.66. 

25.  594,640,000. 

37. 

0.0066062. 

49. 

4.9341. 

13. 

6.696. 

26.  0.00067656. 

38. 

0.65163. 

Exercise 

22. 

Page  50 

1. 

10. 

9.  66.  17.  12,000. 

25.  603.9. 

33.  210. 

2. 

24. 

10.  18.  18.  18,000. 

26.  1282.8. 

34.  945. 

3. 

15. 

11.  100.  19.  660,000. 

27.  184,670. 

35.  5005. 

4. 

35. 

12.  2400.  20.  180,000. 

28.  11,099. 

36.  38,645. 

5. 

8. 

13.  1500.  21.  1034.6. 

29.  1609.9. 

37.  627,400 

6. 

21. 

14.  3500.  22.  2192.3. 

30.  17,458. 

38.  276.67, 

7. 

12. 

15.  8000.  23.  13.31. 

31.  18.212  in. 

8. 

18. 

16.  21,000.  24.  20.265. 

32.  113.04  ft. 

ANSWERS 


7 


Exercise  23.  Page  51 

1. 

7.68964. 

7. 

4.03939. 

13.  0.1248. 

19.  0.02240. 

25. 

22.936, 

&. 

3.68964. 

8. 

2.00010. 

14.  0.0001248. 

20.  0.00015725. 

26. 

34.108, 

3. 

7.68964. 

9. 

1.99999. 

15.  0.0043707. 

21.  1.3020. 

27. 

16.51. 

4. 

3.09497. 

10. 

0.00000. 

16.  0.11422. 

22.  38.079. 

5. 

0.00000. 

11. 

1,248,000. 

17.  0.0000003125. 

23.  3309.6. 

6. 

1.99999. 

12. 

124.8. 

18.  0.25121. 

24.  452.27. 

Exercise  24.  Page  53 

1.  1.97519. 

13.  3.89100. 

25.  5. 

37.  0.00999. 

49. 

60.87. 

2.  3.66078. 

14.  2.00000. 

26.  84. 

38.  0.0709. 

50. 

0.6527. 

3.  1.68618. 

15.  2.11220. 

27.  82.002. 

39.  0.0204. 

51. 

20. 

4.  3.70404. 

16.  2.00286. 

28.  76. 

40.  0.065. 

52. 

50. 

5.  5.00000. 

17.  1.71172. 

29.  35.6. 

41.  0.48001. 

53. 

700. 

6.  9.70000. 

18.  5. 

30.  73.002. 

42.  2.143. 

54. 

800. 

7.  7.00000. 

19.  5. 

31.  92. 

43.  0.4667. 

55. 

9000. 

8.  7.00000. 

20.  3. 

32.  105. 

44.  0.004667. 

56. 

11,000. 

9.  3.76439. 

21.  4. 

33.  63. 

45.  1.913. 

57. 

120,000. 

10.  2.00000. 

22.  3. 

34.  77. 

46.  1.123. 

58. 

0.01. 

11.  2.90000. 

23.  5. 

35.  0.0129. 

47.  12.86. 

59. 

871.1 ; 2, 

12.  6.90000. 

24.  3. 

36.  1290. 

48.  5.184. 

Exercise  25.  Page  54 

1. 

2.60206. 

5.  4.42585. 

9.  0.30103. 

13. 

1.52187. 

17. 

1. 

2. 

3.88606. 

6.  3.36927. 

10.  0.14267. 

14. 

2.20698. 

18. 

0.1, 

3. 

2.56225. 

7.  2.28727. 

11.  1.08092. 

15. 

3.22185. 

19. 

0. 

4. 

T.23433. 

8.  1.14188. 

12.  2.13906. 

16. 

4.15490. 

20. 

1. 

Exercise 

26.  Page  55 

J.  1. 

8.  0.44272. 

15.  6.1649. 

22.  105.47. 

2.  6. 

9.  1.7833. 

16.  0.42742. 

23.  3,013,400. 

3.  3. 

10.  1000. 

17.  1.4179. 

24.  0.081528. 

4.  0.5. 

11.  0.092. 

18.  0.031169. 

25.  232.24. 

5.  1. 

12.  1.8. 

19.  40.464. 

26.  0.0000007237. 

6.  2. 

13.  0.01. 

20.  0.14621. 

27.  103.33. 

7.  0.11111. 

14.  0.21. 

21.  2893.2. 

Exercise  27.  Page  56 

1.  4. 

6.  728.98. 

11.  4,782,800. 

16.  83,522. 

2.  8. 

7.  64. 

12.  16,777,000. 

17.  15,625. 

3.  32. 

8.  125. 

13.  19,486,000. 

18.  6,103,600,000. 

4.  1024. 

9.  1. 

14.  11,391,000. 

19.  15,625. 

5.  80.998. 

10.  40,355,000. 

15.  11.391. 

20.  244,140,000. 

8 


PLANE  TRIGONOMETRY 


21. 

16,413,000,000,000,000. 

29. 

0.0.5765. 

37.  0.023551. 

22. 

7,700,500. 

30. 

0.00000011765. 

38.  0.0001.5228. 

23. 

31,137,000,000. 

31. 

0.018741. 

39.  0.000007.5624. 

24. 

292,360,000,000,000. 

32. 

154.85. 

40.  0.00000012603. 

25. 

2.1435. 

33. 

157.5. 

41.  9.8696;  31.006. 

26. 

180.11. 

34. 

41,961. 

42.  21.991 ; 153.94; 

27. 

0.000000000001. 

35. 

2.0727. 

3053.6. 

28. 

0.00000002048. 

36. 

0.-0019720. 

Exercise 

28.  Page  57 

1. 

1.4142.  7.  5.6569. 

13.  0.54773. 

19.  3.9095. 

2. 

1.71.  8.  3.0403. 

14.  0.3684. 

20.  0.0028827. 

3. 

1.3205.  9.  3.3166. 

15.  0.067405. 

21.  1.7725;  1.4645. 

4. 

1.2394.  10.  1.4422. 

16.  0.064491. 

22.  1.3313;  2.1450; 

5. 

1.1487.  11.  2.802. 

17.  20.729. 

5-5684;  0.42378; 

6. 

2.2795.  12.  1.2023. 

18.  1.9733. 

0.40020;  0.79537. 

Exercise 

29.  Page  59 

1. 

X = 3.  6.  X = 4.2479. 

11.  X = 3. 

16.  X = 3,  y = 1. 

2. 

X = 4.  7.  X = 3.9300. 

12.  X = 3.3219. 

17.  X = 5,  2/  = 1. 

3. 

X = 4.  8.  X = 4.2920. 

13.  X = - 0.087515. 

18.  x = l,y=  1. 

4. 

X = 4.  9.  X = 5.6610. 

14.  X = 4.4190. 

19.  X = 2,  y = 2. 

5. 

X = 3.  10.  X = 3.0499. 

15.  X = - 0.047954. 

20.  x = 3,  y = 2. 

21. 

X = 2,  ?/  = 2. 

27.  X = 2,  - 1. 

35.  2;  7.2730; 

22.  X: 

23.  X: 

24.  X = 

25.  X = 

26.  X = 


log  a — logp 
log(l  + r) 
log  r + log  I — log  a 
log  r 

:1,  -3. 
log  a — log  p 
log(l  + r£) 

log  [s  (r  — 1)  + a]  — log  a 


28.  0.062457. 

29.  3.1389. 

30.  0.036161. 

31.  0.03475. 

6. 

log& 
logo 
logw 
log  5 


2.0009;  2.0043. 


36.  1; 


log  a 


loi 


32. 

33. 

34. 


37. 

38. 


X = 


lb’ 
log  6 


1;3;4 


log  a — log  6 


-1. 


Exercise  30.  Page  62 


1. 

9.65705  - 10. 

13. 

8.89464- 

10. 

25. 

9.95340  - 10. 

37. 

8.11503- 

10. 

2. 

9.97015-  10. 

14. 

9.99651  - 

10. 

26. 

11.13737-  10. 

38. 

8.00469  - 

10. 

3. 

9.90796  - 10. 

15. 

9.23510  - 

10. 

27. 

9.74766  - 10. 

39. 

8.24915  - 

10. 

4. 

9.82551  - 10. 

16. 

9.87099  - 

10. 

28. 

9.66368  - 10. 

40. 

8.24915- 

10. 

5. 

10.57195  - 10. 

17. 

9.68826  - 

10. 

29. 

10.17675-  10. 

41. 

8.63254- 

10. 

6. 

9.32747-  10. 

18. 

10.10706  - 

- 10. 

30. 

9.82332  - 10. 

42. 

8.63205  - 

10. 

7. 

10.57195-  10. 

19. 

9.55763  - 

10. 

31. 

6.51165-10. 

43. 

9.32507- 

10. 

8. 

9.32747  - 10. 

20. 

9.96966  - 

10. 

32. 

8.25667-  10. 

44. 

9.32507  - 

10. 

9. 

9.20613-  10. 

21. 

9.98436  - 

10. 

33. 

6.79257  - 10. 

45. 

10.39604  - 

-10 

10. 

9.99526  - 10. 

22. 

9.42095  - 

10. 

34. 

8.56813  - 10. 

46. 

.7°  30'. 

11. 

9.14412-10. 

23. 

9.48632  - 

10. 

35. 

7.45643  - 10. 

47. 

32°  21'. 

12. 

9.14412-10. 

24. 

9.68916- 

10. 

36. 

8.15611  - 10. 

48. 

58°  27. 

ANSWERS 


9 


49. 

86°  30'. 

55. 

63° 

41' 

23". 

61. 

49°  34 

' 12". 

67. 

57°  4f". 

SO. 

4°3(K. 

56. 

77° 

6'. 

62. 

61°  47 

' 36". 

68. 

49°  25'  7". 

51. 

31°  33'. 

57. 

79°. 

63. 

CO 

O 

00 

48". 

69. 

38°  22'  30' 

52. 

58°  35'., 

58. 

70°. 

64. 

50°  48 

1'  15". 

70. 

2°  3'  30". 

63. 

50°  32'. 

59. 

20° 

13' 

30". 

65. 

O 

00 

30". 

71. 

89°  49'  10' 

54. 

39°  2'. 

60. 

O 

CO 

22' 

15". 

66. 

0 

00 

CO 

o 

Exercise  31.  Page  67 


1.  A = 30°,  B = 60°, 

6 = 10.39, 

S=  31.18. 

2.  B=  30°,  a = 6.928, 

c = 8, 

5 = 13.86. 

3.5=60°,  6 = 5.196, 

c = 6, 

5=  7.794. 

4.  A = 45°,  5 = 45°, 

c = 5.657, 

5 = 8. 

5.  A = 43°  47',  5 = 46°  13', 

6 = 2.086, 

5 = 2.086. 

6.  5 = 66°  30'.  a = 250, 

6 = 575, 

5 = 71,880. 

7.  5 = 61°  55',  a = 1073, 

6 = 2012, 

5=1,079,500. 

8.  5 = 50°  26',  a = 45.96, 

6 = 55.62, 

5 = 1278. 

9.  5 = 54°,  a = 0.5878, 

6 = 0.8090, 

5 = 0.2378. 

10.  A = 68°  13',  a = 185.7, 

6 = 74.22, 

5 = 6892. 

11.  A = 13°  35',  a = 21.94, 

6 = 90.79, 

5 = 995.8. 

12.  5=  85°  25',  6 = 7946, 

c = 7972, 

5 = 2,531,000. 

13.  5=  53°  16',  6 = 65.03, 

c = 81.14, 

5=  1578. 

14.  5 = 4°,  6 = 0.0005594, 

c = 0.00802, 

5 = 0.000002238. 

15.  A = 46°  12',  a = 53.12, 

c = 73.60, 

5 = 1353. 

16.  A = 86°  22',  a = 31.50, 

c = 31.56, 

5=  31.50. 

17.  A = 13°  41',  6 = 4075, 

c = 4194, 

5 = 2,021,000. 

18.  A = 21°  8',  6 = 188.9, 

c = 202.5, 

5 = 6893. 

19.  A = 44° 35',  6 = 2.221, 

c = 3.119, 

5=  2.431. 

20.  5 =52°  4',  a = 3.118, 

c = 5.071, 

5=  6.235. 

21.  A = 31°  24',  5 = 58°  36', 

6 = 7333, 

5 = 16,410,000. 

22.  A = 56°  3',  5 = 33°  57', 

6 = 48.32, 

5 = 1734. 

23.  A = 65°  14',  5 = 24°  46', 

6 = 3.917, 

5=16.63. 

24.  A = 53°  15',  5 = 36°  45', 

a = 1758, 

5 = 1,154,000. 

25.  A = 53°  31',  5 = 36°  29', 

a = 24.68, 

5=  225.2. 

26.  A = 63°,  5 = 27°, 

c = 43, 

5 = 373.9. 

27.  A = 4°  42',  5 = 85°  18', 

c = 15. 

5=  9.187. 

28.  A = 81°  30',  5=8°  30', 

c = 419.9, 

5 = 12,890. 

29.  A = 38°  59',  5 = 51°  1', 

c = 21.76, 

5=115.8. 

30.  A = 1°  22',  5 = 88°  38', 

b = 91.89, 

5 = 100.6. 

31.  A = 39°  48',  5 = 50°  12', 

c = 7.811, 

5=  16. 

32.  A = 30'  12",  5 = 89°  29'  48", 

1 6 = 70, 

5=  21.53. 

33.  A = 43°  20',  5 = 46°  40', 

a - 1.189, 

5 = 0.7488. 

34.  5=  71°  46',  6 = 21.25, 

c = 22.37, 

5 = 74.37. 

35.  5 = 60°  52',  a = 6.688, 

c = 13.74, 

5=40.13. 

36.  5 = 20°  6',  a = 63.86, 

b = 23.37, 

5=  746.15. 

37.  A = 45°  56',  a = 19.40, 

b = 18.78, 

5=182.16. 

38.  A = 41°  11',  6 = 53.72, 

c = 71.38, 

5 = 1262.4. 

39.  A = 55°  16',  a = 12.98, 

c = 15.80, 

5 = 58.42. 

40.  A = 3°  56',  a = 0.5805, 

b = 8.442, 

5 = 2.450. 

10 


PLANE  TRIGONOMETRY 


41. 

5 

= 1C2 

sin-i4 

L cosA. 

43.  S=  1 

b^  tan  A. 

42. 

5: 

= ia^ 

cot  A. 

44.  S=i 

a Vc^  — 1 

2^. 

45. 

A 

= 40° 

45'  48",  B = 49° 

14' 

12",  b = 

:11.6,  C 

= 15.315 

46. 

A 

= 55° 

13'  20",  B = 34° 

'46' 

40",  a = 

: 7.2,  C 

= 8.766. 

47. 

B 

= 61° 

> 

a = 3.647, 

6 = 

■ 6.58,  c 

= 7.523. 

48. 

A 

= 27° 

2'  30 

",  B = 62° 

57' 

30",  a = 

: 10.002,  b 

= 19.595 

49. 

19°  28'  17" 

; 70°  31' 43 

51.  15 

1.498  mi. 

50. 

3112  mi. 

; 19,553 

mi. 

52.  Between  1° 

15'  30"  and  1°19'  10 ", 

53. 

212.1  ft. 

58. 

59°  44'  35". 

63. 

7.071  mi. 

; 

67. 

685.9  ft. 

54. 

732.2  ft. 

59. 

95.34  ft. 

7.071  mi. 

68. 

5.657  ft 

55. 

3270  ft. 

60. 

23°  50'  40". 

64. 

19.05  ft. 

69. 

136.6  ft. 

56. 

37.3  ft. 

61. 

36°  1'  42". 

65. 

20.88  ft. 

70. 

140  ft. 

57. 

1°  25'  56' 

62. 

69°  26'  38". 

66. 

56.65  ft. 

71. 

84.74  ft. 

Exercise 

32 

. Page 

71 

1. 

. C = 

2(90' 

°-A),  c = 

2 a 

cos  A,  A 

= a sin  A 

2, 

, A = 

90°  - 

-iC,  c = 

2a 

cos  A,  A 

= a sin  A 

3.  C = 2(90°  — A),  a = — ^ , ^ = asinA. 

2 cos  A 


4.  A = 90°— iC,  a = — - — , A r=  a sin  A. 

^ 2 cos  A 

6.  C = 2(90°- A),  a=z— c = 2acosA. 
sin  A 

6.  A = 90°  — ,T  (7,  a — — ~ — , c = 2 a cos  A . 

. ■ sin  A 

7.  sinA  = -,  C = 2(90°  — A),  c = 2acosA. 

8.  tanA  = — , C = 2(90°-A),  a = — — 

c sin  A 

9.  A = 67°  22'  50",  C = 45°  14'  20",  h = 13.2. 

10.  c = 0.21943,  h = 0.27384,  S = 0.03004. 

11.  a = 2.05.5,  h = 1.6852,  S = 1.9819. 

12.  a = 7.706,  c = 3.6676,  S = 13.725. 

13.  A = 25°  27' 47",  0 = 129°  4' 26",  a = 81.41,  A = 35. 

14.  A = 81°  12'  9",  C = 17°  35'  42",  a = 17,  c = 5.2. 

15.  c = 14.049,  A = 26.649,  5 = 187.2. 

16.  S = a2sin^OcosiC.  19.28.284  ft.;  21.  94°  20'.  24.  37.699  sq.  in. 

17.  S = sin  A cos  A.  4525.44  sq.  ft.  22.  2.7261.  25.  0.8775. 

18.  5 = A^tan^^C.  20.0.76536.  23.  38°  56' 33". 

Exercise  33.  Page  72 

1.  r=  1.618,  A = 1.5388,  5 = 7.694.  4.  r = 1.0824,  c = 0-82842,  5 = 3.3137. 

2.  A = 0.9848,  p = 6.2514,  5 = 3.0782.  5.  r = 2.5942,  A = 2.4891,  c = 1 .461. 

3.  A = 19.754,  c = 6.257,  5 = 1236.  6.  r = 1.5994,  A = 1.441,  p = 9.716. 

7.  0.51764  in.  9.  0.2238  sq.  in.  13.6.283. 

g _ c 10.  0.310  in.  14.  0.635  sq.  in 

~ 90°'  11.  1.0235  in. 

12.  0.062821 ; 6.2821. 


ANSWERS 


11 


Exercise  34. 

Page  73 

2.  29.76  sq.  in. 

13.  52°  35' 42". 

25.  362.09  ft. 

36.  2675.8  mi. 

3.  104.07  sq.ft. 

14.  60°  36' 58". 

26.  59°  2' 10". 

37.  25.775  ft.; 

4.  36.463  sq.  in. 

15.  6.3509  in. 

27.  14.772  in.  ; 

19.45  ft. 

5.  20.284  in. 

16.  20  in. 

15.595  in. 

38.  10.941ft.; 

7.  37.319  ft. 

17.  7.7942  in. 

28.  73.21  ft. 

20.141  ft. 

8.  342.67  ft. 

18.  40°  7'  6". 

29.  25°  36' 9". 

39.  55.406  ft. 

9.  36.602  ft.; 

19.  77°  8' 31". 

30.  26.613  in. 

40.  Between  131 

86.602  ft. 

20.  94.368  ft.; 

31.  7.5  ft. 

and  132'. 

10.  120.03  ft. 

25°  42'  58". 

32.  59°  58' 54"; 

41.  43°  18' 48". 

11.  2.9101  mi.; 

21.  24.652  ft. 

173.08  ft. 

42.  2.6068  in. 

3.531  mi. 

22.  196.93  ft. 

33.  7.2917  ft. 

43.  14.542  in. ; 

12.  11°  47"; 

23.  220.8  ft. 

34.  19.051. 

26.87  in. 

49.206  ft. 

24.  1915.8  ft. 

35.  1.732  in. 

44.  6471.7  ft. 

Exercise 

35.  Page 

80 

29.  10. 

33.  11. 

37.  0. 

41.  5.10. 

45.  28^  in. 

49.  |V3. 

30.  15. 

34.  3|. 

38.  7. 

42.  5.10. 

46.  9.43  in. 

50.  Yes. 

31.  13. 

35.  3. 

39.  5. 

43.  8.24. 

47.  2. 

51.  Octagon 

32.  21. 

36.  5. 

40.  15. 

44.  4.24. 

48.  3 Vs. 

2.829.' 

Exercise 

36. 

Page  84 

16. 

I. 

18.  II. 

20. 

III. 

22. 

I. 

24.  III. 

26. 

I. 

28.  III. 

17. 

I. 

19.  II. 

21. 

IV. 

23. 

II. 

25.  IV. 

27. 

II. 

29.  On  OF'. 

30. 

On  OX. 

64. 

sin  = i V2  ; cos  = 

— i V2  ; tan  = — 1 ; 

61. 

^V3; 

j 

V6. 

CSC  = V2  ; sec 

= - 

V2  ; 

cot  = — 1. 

62. 

90°. 

65. 

sin  =:  0 ; cos  = 

- 1 

; tan 

= 0 ; 

63. 

60°. 

CSC  = 00  ; sec  = 

1;  cot 

- 00. 

Exercise  37.  Page  88 


62.  2 ; one  in  Quadrant  I,  one  in  Quadrant  II. 

83.  4 ; two  in  Quadrant  I,  two  in  Quadrant  IV. 

54.  2;  1;  1;  1;  1. 

65.  Between  90°  and  270° ; between  0°  and  90°  or  between  180°  and  270® ; 
between  0°  and  90°  or  between  270°  and  360° ; between  180°  and  360°. 


57.  1 ; 0 ; 0 ; c»  ; 

1 ; CO  ; 1 ; 0. 

69.  Ill;  II. 

60.  40;  20. 

61.  0. 

62.  0. 

63.  0. 

64.  4a5, 


65.  - 2(a2  + 62). 

66.  0. 

67.  {'. 

76.  30°  ; 150°  ; 390° ; 510°. 

77.  30°  ; 330°  ; 390°  ; 690°. 

78.  60°  ; 120°  ; 420°  ; 480°. 

79.  60°  ; 300°  ; 420°  ; 660°. 

80.  30°  ; 210°  ; 390°  ; 570°. 


81.  60°  ; 240°  ; 
420°;  600°. 

82.  210°;  330°. 

83.  120°;  240°. 

84.  225°;  315°. 

85.  135°;  225°. 

86.  185°;  315°. 

87.  135°;  315°. 


12 


PLANE  TRIGONOMETRY 


Exercise 

38.  Page  91 

1. 

sin  10°. 

9.  tan  78°. 

17.  - cot  65°. 

25.  - sin  7°  10' 3". 

2. 

— cos  20°. 

10.  cot  82°. 

18.  - cot  1.3°. 

26.  cos  8.5°  54' 46". 

3. 

— tan  32°. 

11.  - sin  85°. 

19.  — sin  0°. 

27.  - tan  37°  51'  4.5" 

4. 

— cot  24°. 

12.  — sin  15°. 

20.  COS  0°. 

28.  cotl5°10'3". 

5. 

sin  0°. 

13.  — tan  78°. 

21.  sin  31°  50'. 

29.  sin  32.2.5°. 

6. 

— tan  0°. 

14.  — tan  35° 

22.  — COS  12°  20'. 

30.  — cos  52.25°. 

7. 

— sin  20°. 

15.  cos  70°. 

23.  tan  85°  30'. 

8. 

— cos  45°. 

16.  cos  10°. 

24.  - cot  72°  20'. 

Exercise 

39.  Page  93 

1. 

cos  10°. 

10.  — cot  9°. 

19.  - sin  86°. 

28.  - cot  9.1°. 

2. 

cos  30°. 

11.  - cot  29°. 

20.  cos  75°. 

29,  0.0262. 

3. 

cos  20°. 

12.  - cot  39°. 

21.  cos  87°. 

30.  - 0.5483. 

4. 

cos  40°. 

13.  - tan  4°  V. 

22.  — sin  5°. 

31.  - 0.7729. 

5. 

- sin  5°. 

14.  — tan  7°  2'. 

23.  tan  80°. 

32.  0.5040. 

6. 

— sin  7°. 

15.  - tan  8°  3'. 

24.  tan  30°. 

33.  - 0.1304. 

7. 

— sin  21°. 

16.  - tan  9°  9'. 

25.  — tan  20°. 

34.  0.8686. 

8. 

— sin  37°. 

17.  - sin  3°. 

26.  - cot  1.5°. 

35.  0.1357. 

9. 

— cot  1°. 

18.  — sin  9°. 

27.  - cot  7.8°. 

36.  - 0.1354. 

37. 

9.89947  - 10. 

40.  -(10.52286-10).  43. 

10.147-53  - 10. 

38. 

- (9.83861  - 

10).  41.  -(9.91969-10).  44. 

-(9.82489-  10). 

39. 

-(9.79916- 

10).  42.  9.92401 

- 10.  46. 

225°;  315°;  585°;  675° 

Exercise 

40.  Page  95 

6. 

.«?in  T.  — -1- 

1 

19.  45°. 

27.  60°. 

V COt^  X + 1 

20.  30°. 

28.  60°  or  180°. 

7. 

ms  r.  — -J- 

1 

21.  60°. 

29.  4-5°. 

V tan'^  X + 1 

22.  45°. 

30.  30°. 

8. 

.SPP.  O'.  — 4- 

1 

23.  45°. 

31.  4-5°. 

Vi 

— sin^  X 

24.  45°. 

32.  4 V5  ; 4 V5. 

9. 

CSC  X 4- 

1 

25.  60°. 

33.  lVl5;Vl5 

vr 

— COS^  X 

26.  45°. 

34.  4;  5. 

35. 

36. 

37. 

38. 

39. 

40. 

53. 

54. 
65. 

56. 

57. 

58. 

59. 

60. 


= I Vs,  cosx  = ^ Vs,  tanx  = 2 ; cscx  = ^ Vs,  secx  = Vs,  cot  x = 


45.  270°  or  SO'’. 

46.  30°  or  150° 

47.  45°,  135°,  225°, 
or  315°. 

48.  60°. 


t'LVT7;  ^iyVl7.  41.  45°  or  225°. 

,4;  42.  45°,  135°,  225°, 

When  X = 0°.  or  315°. 

0°  or  180°.  43.  45°  or  225°. 

38°  10'.  U.  0°  or  60°. 

cosA=^-V5,  tanA=|-V5,  cscA=  |,  _ secA=:|-V5,  cotJ.=  i V5. 

sinA=lV7,  tanA=^V7,  cscA=iV7,  secA=^,  cotA=  SV7. 

sinA=  -y^jVlS,  cosA  = Vl3,  cscA=  ;jVl3,  secA  = |Vl3,  cotA= 
sinA=^,  cosA=3,  tanA=^,  _ cscA  = |,  secA  = §. 

sinA=^V5,  cosA=^,  tanA  = i^Vs,  cscA  = |Vs,  cotA=|%5- 

oosA  = tan  A = esc  A = -{-f , sec  A = -G-,  cot  A = 

cosA  tan  A = f , esc  A = |,  sec  A — cot  A = |. 

sinA  = tan  A = cscA  = sec  A = cot  A = 


answers 


13 


5 A = -2^,  cot  A = 


61.  sin  A = If,  tan  A = csc^  = 

62.  sin^  = i,  cos^=|,  cscJ.  = |,  sec^  = cot^ 

63.  sin  A = V2,  cos  A = I V2,  tan  Jl  = 1,  esc  A = V2,_  sec  A = V2 . 

64.  sin  A = i V5,  cos  A = I VB,  tan  A = 

65.  sin^  = Vs,  cos  tan  4 = 

66.  sin  ^ i Vi,  cos  A = ^ Vi,  tan  A = 

67.  cosJ.  = Vl—  m?,  tan.4  ^ 


2,  CSC  .<4  = ^ Vs,  sec.4  = V6._ 
vi,  csc^  = f Vi,  cot  5 Vi. 

1,  sec^  = V2,  cot^  = l. 

2m 


Vl- 


y COt^  = 


1 .1  1 
CSC  A = —,  sec  A = — --- 

m Vl— m2 

70.  cos  0°  = 1,  tan  0°  = 0,  esc  0“^  = oo, 

71.  cos  90°  = 0,  tan  90°  = oo,  esc  90°  = 1, 

72.  sin  90°  = 1,  cos  90°  = 0,  esc  90°  = 1, 

73.  sin  22°  30'  = — ^ , cos  22°  30' 

V4  + 2V2 

CSC  22°  30'  = V4  + 2 Vi,  sec  22°  30' 

1 — cos2  A 

74.  [■ 

cos  A 


Vl— 5 


68. 


69. 


1 — m2 
m2  + n2 


m 2 mn 

sec  0°  = 1,  cot  0°  = 00. 
sec  90°  = cx),  cot  90°  = 0. 
sec  90°  = 00,  cot  90°  = 0. 

, tan  22°  30'  = V2  — 1, 


1 


V4-  2 V2 
y/i-  2 Vi. 


COs2.4 
1 — cos2  A 


Exercise 

41. 

Page 

98 

1. 

0.25875. 

5. 

1. 

9.  O.i 

366. 

13. 

0.5. 

2. 

0.96575. 

6. 

0. 

10.  - 

0.5. 

14. 

- 0.866. 

3. 

0.96575. 

7. 

0.96575. 

11.  0. 

707. 

15. 

0.25875.^ 

4. 

0.25875. 

8. 

- 0 

.25875. 

12.  - 

0.707 

16. 

- 0.96575 

Exercise 

42. 

Page 

99 

1. 

0.268. 

5. 

CO. 

9.  - 

1.732 

13. 

- 0.577. 

2. 

3.732. 

6. 

0. 

10.  - 

0.577 

14. 

- 1.732. 

3. 

3.732. 

7. 

- 3 

.732. 

11.  - 

1. 

15. 

- 0.268. 

4. 

0.268. 

8. 

- 0 

.268. 

12.  - 

1. 

16. 

- 3.732. 

Exercise 

43. 

Page 

102 

1. 

tl- 

14. 

— COS  y. 

27. 

1 — tan  y 

2. 

H- 

15. 

— sin  y 

1 + tan  y 

3. 

3 3 
^5* 

16. 

sin  y. 

28. 

Vs  cot  y - 

- 1 

4. 

fl- 

17. 

sin  X, 

cot  y 

+ Vs 

5. 

111- 

18. 

— cos  X. 

29. 

^ Vi  cot  1/  + 1 

6. 

M- 

19. 

— sin  X. 

cot  2/ 

„ 1 

Vi 

7. 

COS  y. 

20. 

— cotx. 

30. 

tanw. 

8. 

sin  y. 

21. 

tan  X. 

31. 

0.8571 ; 0 

.2222. 

9. 

coty. 

22. 

— tanx. 

32. 

3.732 

; 0.268. 

10. 

COS  y. 

23. 

cotx. 

33. 

1 . 4 9 
^ y 7 I 

; 45°. 

11. 

sin  y. 

24. 

— sin  y 

34. 

x+y 

= 90°,  270°  in 

12. 

— sin  y. 

25. 

^■V2(cos2/  — 

sin  y) . 

the 

three  cases. 

13. 

— cosy. 

26. 

Vi  (cos  y + 

sinw). 

37. 

135°, 

405° 

14 


PLANE  TRIGONOMETRY 


5.  1.  - 

6.  > Vs.  8. 


Exercise  44.  Page  103 
9.  0.8492.  11.  - 1.1776.  13. 

10.  0.5827.  12.  1.7161.  14.  -Vf 


15.  3 sin  a:  — 4 sin®a;. 

16.  4 cos^i  — 3 cos  X. 


Exercise  45.  Page  104 

1.  0.2588.  3.  0.2679.  5.  7.5928.  7.  0.9239.  9.  2.4142. 

2.  0.9659.  4.  3.7321.  6.  0.3827.  8.  0.4142.  10.  5.0280. 

11.  0.10051;  0.99493.  12.  0.38730;  0.92196  ; 0.42009;  2.3805. 


8. 

9. 

15. 


0. 

iV3. 

2 


sin  2x 
16.  2 cot2x. 
cos  (x  — y) 


17. 


sin  X cos  y 


Exercise  46.  Page  105 


18. 


cos  (x  + y) 
sin  X cos  y 
19.  tan^x. 
cos  (x  — y) 


20. 


21. 


cos  X cos  y 
cos  (x  4-  y) 
cos  X cos  y 


gg  cos(x-y) 
sin  X sin  y 
cos(x  + y) 
sin  X sin  y 
24.  tan  x tan  y. 
27. 


Exercise  47.  Page  109 

1.  a = 6 sin  A ; 6 = a sin  R ; a = b-,  sin  A = sin  R. 

4.  Sin. 

5.  1000  ft. 

Exercise  48 


1.  C = 123°  12',  b = 2061.6,  c = 2362.6. 

2.  G = 66°  20',  b = 667.^9,  c = 663.99. 

3.  C = 36°  4',  b = 677.31,  c = 468.93. 

4.  G=26°12',  6 = 2276.6,  c = 1673.9. 

5.  C = 47°  14',  a = 1340.6,  b - 1113.8. 

6.  A = 108°  60',  a = 63.276,  c = 47.324. 

7.  R = 66°  66',  b = 5685.9,  c = 5357.5. 

8.  R = 77°,  a = 630.77,  c = 929.48. 

9.  a = 5 ; c = 9.659. 

10.  a = 7;  6 = 8.573. 

19.  8 and  5.4723. 

20.  4.6064  mi. ; 4.4494  mi.;  3.7733  mi. 

21.  5.4709  mi. ; 5.8013  mi. ; 4.3111  mi. 


6.  8.5450  in.;  4.2728  in. 

7.  27.6498  in. 

8.  9.1121  in. 

Page  110 

11.  Sides,  600  ft.  and  1039.2  ft. ; 
altitude,  519.6  ft. 

12.  855:  1607. 

13.  5.438;  6.857. 

14.  15.588  in. 

15.  AR=  59.564 mi.; 

AG  = 54.285  mi. 

16.  4.1365  and  8.6416. 

17.  6.1433  mi.  and  8.7918  mi. 

18.  6.4343  mi.  and  5.7673  mi. 


Exercise  50 

1.  Two.  3.  No  solution. 

2.  One.  4.  One. 

9.  R = 12°  13'  34' 

10.  R = 57°  23'  40' 

11.  R = 41°  12' 56' 

12.  A = 54°  31', 

13.  R = 24°  57' 26 
R'=  155°  2'  34' 


Page  115 

5.  One. 

6.  Two. 

G = 146°  15'  26",  c = 1272.1. 
G = 2°  1'  20",  c = 0.38525. 
G = 87°  38'  4",  c = 116.83. 
G = 47°  45',  c = 50.496. 


7.  No  solution. 

8.  One. 


G = 133°  48'  34",  c = 615.7  ; 
G'=  3°  43'  26",  c'=  55.414. 


ANSWERS 


15 


14.  A = 51°  18' 27", 
A'=  128°  41' 33" 

15.  A = 147°  27'  47" 
A'=  0°  54'  13", 

16.  5 =44°!' 28", 
B'=  135°  58'  32" 

17.  .B=90°, 

18.  420.  19. 


C = 98°  21'  33" 
C'=  20°  58' 27". 
B = 16°  43'  13" 
B'=  163°  16' 47' 
C = 97°  44'  20", 
C'=  5°  47'  16", 
C = 32°  22'  43", 
124.62. 


, c = 43.098; 
, c'=  15.593. 

, a = 35.519; 
',  a'=  1.0415. 

, c = 13.954 ; 
c'=  1.4202. 
c = 2.7901. 

20.  3.2096  in. 


21.  AS  =3.8771  in.;  BC  = 2.3716  in. ; CD  = 3.7465  in. ; AD  = 6.1817  in. 

22.  C = 125°6',  J)=  93°24';  AB  = 4.3075  in. ; BC  = 3.1288  in.;  CB  = 5.431  in. ; 
BE  = 4.4186  in. ; AE  = 5.0522  in. 


Exercise  51.  Page  117 


2.  b = a cos  C + c cosA  ; „ , 6-  + — a- 

r ^ , T,  °.  cosA  = ; , 

a = 0 COSU  + c cosB;  2 6c 

c = 6cosA.  14.  AC  = 8.499  in. ; 

4.  Impossible.  BB  = 3.1254  in. 

5.  5.  15.  BC=  5.9924 in.; 

6.  7.655.  BB=  8.3556  in. 

7.  7. 


90° 


16.  AB  = 1.9249  in. ; 
CB  = 4.4431  in. ; 

A = 109°  26'; 

B = 112°  13' 40" 
C = 88°  11' 40"; 
B = 50°8'40". 

17.  13.3157  in. 


Exercise  52. 


a — b 


1. 

a + 6 

2.  tan  J (A 

3.  a = b. 


tan  (A  - 45°). 
B)  = 0. 


4.  a + 6 = (ct  — 6)  (2  + Vs). 


Page  119 
0_  0 

13.  ^ = «,V3. 

0 

14.  tan  ^ (A  — B)  = 0 ; A = B. 


11. 

2 sin 

A 

tan  A 

17.  5. 

— . 

or  00  = CO. 

0 

0 

18.  SidesAB,  BC, 

AE ; diagonal  AB ; 

angles  B. 

, CAB 

, BAE. 

Exercise  53.  Page  121 

1.  A 

= 51°  15',  B 

= 56°  30', 

c 

= 95.24. 

2.  B 

= 60°  45'  2",  C 

= 39°  14'  58", 

, a 

= 984. 

83. 

3.  A 

= 77°  12'  53",  B 

= 43°  30'  7", 

c 

= 14.987. 

4.  B 

= 93°  28'  36",  C 

= 50°  38'  24", 

a 

= 1.3131. 

5.  A 

= 132°  18' 27",  B 

= 14°  34'  24", 

, C 

= 0.67 

75. 

6.  A 

= 118°  55'  49",  C 

= 4.5°  41'  3.5", 

b 

= 4.1.554. 

7. 

B 

= 

65° 

13'  51", 

C=  28°  42'  5", 

a = 3297.2. 

19. 

6. 

8. 

A 

= 

68° 

29'  15", 

B = 45°  24'  18", 

c = 4449. 

20. 

10.392. 

9. 

A 

117 

° 24'  32", 

, B=  32°  11' 28", 

c = 31.431. 

21. 

A = B = 90°  - J C, 

10. 

A 

= 

2°  46'  8", 

B = 1°  54'  42", 

c = 81.066. 

a sin  C 

11. 

A 

116°  33'  54", 

B = 26°  33'  54", 

c = 140.87. 

sinA 

12. 

A 

:= 

6°1 

' 55", 

B = 108°  58'  5", 

c =862.5. 

22. 

8.9212. 

13. 

A 

= 

45° 

14'  20", 

B=  17°  3' 40", 

c = 510.02. 

23. 

25. 

14. 

A 

= 

41° 

42'  33", 

B = 32°  31'  1.5", 

c = 9.0398. 

24. 

3800  yd. 

15. 

A 

izr 

62° 

58'  26", 

B = 21°  9'  58", 

c = 4151.7. 

25. 

729.67  yd.  ^ 

16. 

A 

= 

84° 

49'  58", 

B = 28°  48'  26", 

c = 42.374. 

26. 

430.85  yd. 

17. 

B 

— 

24° 

ir  20", 

C = 144°  55'  52". 

, a = 205. 

27. 

10.266  mi. 

18. 

B 

= 

20° 

36'  34", 

C=  102°  10' 14", 

, a -37.5. 

28. 

2.3385  and  5.0032. 

16 


PLANE  TRIGONOMETRY 


Exercise  54.  Page  125 

1.  ^[log(s  — 6)+ log(s— c)+ cologs+ colog(s— a)].  4.  log r + colog (s  — a) . 


2. 

i[log(s- 

b)  + log(s  — 

c)  + colog  b + 1 

colog  c]. 

5.  log(s 

— a)  + log  taniA, 

3. 

^[log(s- 

a)  + log(s  — 

b)  + log  (s  — c) 

+ cologs].  6.  The  second. 

7.  Vj,  or  0.37796 

; 41°  24' 34". 

9. 

A = 60°. 

Exercise  55. 

Page  : 

127 

1. 

38°  52'  48" 

';  126°  52' 12 

";  14°  15'. 

17. 

45°;  120°; 

15°. 

2. 

32°  10'  65" 

■;  136°  23' 50 

";  11°  25' 15". 

18. 

45°  ; 60°  ; 7 

■5°. 

3. 

27°  20'  32' 

';  143°  7' 48" 

; 9°  .31' 40". 

19. 

84°  14'  34". 

4. 

42°  6'  13" ; 

; 56°  6' 36"; 

81°  47' 11". 

20. 

54°  48'  54". 

5. 

16°  25'  36" 

';  30°  24';  133°  10' 24". 

21. 

10.5°;  15°; 

60°. 

6. 

46°  49'  35" 

';  57°  59' 44" 

; 75°  10' 41". 

22. 

54.516. 

7. 

26°  29";  43°  25' 20";  110°  34' 11". 

23. 

O * 

O 

8. 

49°  34'  58" 

';  58°  46' 58" 

; 71°  38' 4". 

24. 

12.434  in. 

9. 

51°  53'  12" 

' ; 59°  31'  48" 

; 68°  35'. 

25. 

4°  23'  2"  W 

. of  N.  or  W.  of  S. 

10. 

36°  52'  12" 

■;  53°  7' 48"; 

90°. 

26. 

A = 90°  37' 

3"; 

11. 

36°  52'  12" 

• ; 53°  7'  48"  ; 

: 90°. 

B = 104°  28 

;'  41"; 

12. 

33°  33'  27" 

';  33°  33' 27" 

';  112°  53' 6". 

C = 96°  55' 

44"; 

13. 

60°;  60°; 

60°. 

D = 67°  58' 

32". 

14. 

28°  57'  18" 

';  46°  34' 6"; 

104°  28' 36". 

27. 

82°  49'  10". 

15. 

36°  52'  12" 

';  53°  7' 48"; 

90°. 

28. 

36°  52'  11"; 

16. 

8°  19'  9" ; 

33°  33'  36" ; 

1—* 

CO 

00 

0 

53°  7'  49". 

1.  277.68. 

2.  452.87. 

3.  8.0824. 


Exercise  56.  Page  128 

4.  27.891.  7.  10,280.9. 


5.  139.53. 

6.  1380.7. 


8.  82,362. 

9.  409.63. 


Exercise  57.  Page  129 


10.  1,067,750. 

12.  10.0067  sq.  in. 

13.  18.064  sq.  in./ 

14.  13.41  sq.  in. 


1. 

85.926. 

3. 

436,540. 

5.  7,408,200. 

7.  1 

.76,384.  9. 

92.963. 

2. 

23,531. 

4. 

157.63. 

6.  398,710. 

8.  25,848.  10. 

3176.7. 

11. 

5.729  sq.  in. 

Exercise  58.  Page  131 

1. 

6. 

14. 

8160. 

29. 

13.93  ch.,  23.21  ch.,  32.50  ch. 

2. 

150. 

15. 

26,208. 

30. 

14  A.  5.54  sq. 

cll. 

3. 

43.301. 

16. 

17.3206. 

31. 

30°  ; 30°  ; 120°. 

4. 

1.1367. 

17. 

10.392. 

32. 

2,421,000  sq. : 

ft. 

5. 

10.279. 

18. 

365.68. 

33. 

199  A.  8 sq.  ch. 

6. 

16.307. 

19. 

29,450;  6982.8. 

34. 

210  A.  9.1  sq. 

ch. 

7. 

1224.8  sq. 

rd. ; 

20. 

15,540. 

35. 

12  A.  9.78  sq. 

ch. 

7.655  A. 

21. 

4,333,600. 

37. 

876.34  sq.  ft. 

8. 

3.84. 

22. 

13,260. 

38. 

1229.5  sq.  ft. 

9. 

4.8599. 

24. 

3 A.  0.392  sq.  ch. 

39. 

9 A.  0.055  sq. 

ch. 

10. 

10.14. 

25. 

12  A.  3.45  sq.  ch. 

41. 

1075.3. 

11. 

62.354. 

26. 

4 A.  6.634  sq.  ch. 

42. 

2660.4. 

12. 

0.19975. 

27. 

61  A.  4.97  sq.  ch. 

43. 

16,281. 

13. 

240. 

28. 

4 A.  6.633  sq.  ch. 

45. 

Area  ==  ah  sin 

A. 

ANSWERS 


17 


Exercise  59.  Page  133 


1. 

20  ft. 

13. 

260.21  ft. ; 

25. 

50°  29' 35";  o, 

2. 

37°  34'  5". 

3690.3  ft. 

39°  30'  25".  a + b 

3. 

30°. 

14. 

2922.4  mi. 

26. 

74°  44'  14".  36.  30°. 

4. 

199.70  ft. 

15. 

60°. 

27. 

3.50.61  in.  37.  97.86 in.; 

5. 

106.69  ft. ; 

16. 

3.2068. 

28. 

115.83  in.  153.3  in. ; 

142.85  ft. 

17. 

6.6031. 

29. 

388.62  in.  159.31  in. 

6. 

43.12  ft. 

18. 

2.38,410  mi. 

30. 

83°  37' 40".  38.  1302.5  ft.; 

7. 

78.-36  ft. 

19. 

1..3438  mi. 

31. 

97°liq  33°  6' 51". 

8. 

75  ft. 

20. 

861,860  mi. 

32. 

89°  50' 18".  39.  0.9428. 

9. 

1.4442  mi. 

21. 

235.81  yd. 

33. 

0.2402;  41.45  ft. 

10. 

56.649  ft. 

22. 

26°  34'. 

1.9216  in.;  43.  0.9524. 

11. 

2159.5  ft. 

23. 

69.282  ft. 

33.306  in.  2hVl^  + w^ 

12. 

7912.8  mi. 

24. 

49°  18'  42"  : 

; 34. 

1.7  in. ; — !r  — vfi 

40°  41'  18". 

0.588  in. 

Exercise 

60.  Page  137 

4. 

460.46  ft. 

8. 

422.11yd. 

12. 

255.78  ft.  16.  210.44  ft. 

5. 

88.936  ft. 

9. 

41.411ft. 

13. 

529.49  ft.  18.  19.8;  35.7; 

6. 

56.564  ft. 

10. 

234.51  ft. 

14. 

294.69  ft.  44.5. 

7. 

51.595  ft.  . 

11. 

12,492.6  ft. 

15. 

101.892  ft. 

19. 

13.657  mi.  perjiour. 

- . OB  sin  0 

24.  X — ; 

28.  658.361b. ; 22°  23' 47' 

20. 

N.  76°56'E. ; 

sin  a 

■with  first  force. 

13.938  mi.  per  hour. 

sin  a : 

29.  88.3261b.;  45°  37' 16' 

21. 

3121.1  ft. ; 

90°; 

B = 90° ; 

■with  known  force. 

3633.5  ft. 

Z a = 

: 90°  - 0.  30.  757.50  ft. 

22. 

25.433  mi. 

25.  288.67  ft. 

31.  520.01yd. 

23. 

6.3397  mi. 

26.  11.314 mi.  per  hour.  32.  1366.4ft. 

35.,  536.28  ft. 

; 500.16  ft. 

36.  345.46  yd.  37.  61.23  ft. 

1.  i9,647  sq.  ft. 

2.  27.527  sq.  in. 


Exercise  61.  Page  141 

3.  41.569  sq.  in. 

4.  6. 


6.  I;  iVi. 

9.  6. 

11.  40,320  sq.  ft. 


Exercise  62.  Page  142 


1.  11.124A. 

2.  21.617A. 

3.  15.129A. 


4.  14A. 

5.  13.77A. 

6.  10.026A. 


7.  lOA. 

8.  4.5.348  A. ; 
10.4652A. 


9.  36.38A. 

10.  20.07 A. 

11.  3.766A. 

12.  2.485A. 


Exercise 

1.  0.5223  sq.  in. 

2.  66.2343  sq.  in. 

3.  3.583sq.  in. ; 27.6565  sq.  in. 


63.  Page  144 

4.  8.6965  sq.  in. 
6.  112.26  sq.  in.; 
201.9  sq.  in. 


6.  0.14279. 

7.  116.012  sq.  in. 

8.  |. 


18 


PLANE  TRIGONOMETRY 


Exercise  64.  Page  147 


1. 

18'  23" ; 

5. 

13'  53" ; 

10. 

101.44  mi. 

18.385  mi. 

20.787  mi. 

11. 

11.483  mi.  — 

2. 

37'  29"  ; 

6. 

19'  52" ; 

12. 

44.5  mi..-^ 

37.4775  mi. 

12°  57'  8"  S. 

13. 

S.  75°  31' 20"  E.; 

3. 

51'  33"  ; 

7. 

35.207  mi. 

23.2374  mi. 

34.445  mi. 

8. 

16.6296  mi. ; 

14. 

N.  17°  6'  14"  W. ; 

4. 

37'  16" ; 

11' 6.7". 

32°  .50' 30"  N. 

7.4135  mi. 

9. 

59.155  mi. 

15. 

23.8.54  mi.; 

16. 

27.803  mi.; 

N.  52°  18' 21" 

W. 

S.  56°  58'  34"  E. 

Exercise  65.  Page 

148 

1. 

42°16'N.; 

68°  54'  39"  W. 

2.  103.57  mi. 

3. 

60°15'N.;  62°  15' 55"  W. 

1.  31°  26' 16"  N.; 
41°  44'  23"  W. 

2.  S.63°26'W.; 
42.486  mi. ; 

16°  14' 52"  W. 


Exercise  66.  Page  149 

3.  41°  50' 5"  N.; 

68°  15'  1"  W. 

4.  16.727  mi. ; 

30°  16'  19"  W. 

5.  N.  77°  9'  38"  W. ; 

3:3°  11'  W. 

Exercise  67.  Page  150 


6.  40°  4' 16"  N.; 
72°  44' 56"  W. 

7.  42°  47' 4.3"  N. ; 
70°  48' 25"  W. 


1.  35°  49' 10"  S.;  22°  2' 44"  W.;  N.  61°42'W.;  183.16  mi. 

2.  42°  15'  29"  N. ; 69°  6'  11"  W. ; 44.939  mi. 

3.  32°  53'  34"  S. ; 13°  1'  53"  E ; 287.16  mi. 

4.  41°  1'  40"  N. ; 69°  54'  1"  W. 

5.  57' 19";  21.4  mi. 

6.  1°37'8";  45.652  mi. 

Exercise  68.  Page  152 

1.  fir. 

2. 

3.  ^\lT. 

4. 

25.  216°,  fir. 

26.  300°,  I IT. 

27.  120°,  §7T. 


5.  if-J-TT.  9.  270°.  13. 

6.  8 IT.  10.  240°.  14. 

7.  J^ir.  11.  210°.  15. 

8.  -5/77.  12.  225°.  16. 

28.  33°  45',  IT. 

29.  0.017453; 
0.0002909. 


7°  30'. 

17.  II. 

21. 

n. 

540°. 

18.  II. 

22. 

II. 

1080°. 

19.  III. 

23. 

I. 

1800°. 

20.  IV. 

24. 

III. 

30.  3437.75';  206,265". 

31.  IT  radians. 

32.  ^ IT  radians. 


Exercise  69.  Page  154 

1.  16°,  164°,  376°,  524°. 

2.  30°,  150°,  390°,  610°,  750°,  870°. 

3.  30°,  150°,  390°,  510°,  750°,  870°,  1110°,  1230°. 

4.  67°  30',  112°  30',  427°  30',  472°  30'. 

9.  0.00058177632.  10.  0.000582. 


5.  18°,  162°,  378°,  522°. 

6.  0.99999995769. 

7.  0.00029088820. 

8.  0.00029088821. 

11.  0.0175. 


ANSWERS 


19 


Exercise  70.  Page  155 


1.  60^,  300=’. 

5.  45°,  225°. 

9.  26°  34',  206°  34', 

2.  - 60=,  - 300°. 

6.  _ 135°,  - 315°. 

386°  34',  566°  34'. 

3.  25°,  335°, 

7.  60°,  240°, 

10,  - 116°  34',  - 296°  34', 

385°,  695°. 

420°,  600°. 

- 476°  34',  - 656°  34'. 

4.  60°,  300°, 

8.  30°,  210°, 

420°,  660°. 

390°,  570°. 

Exercise  71,  Page  156 

5.  60°,  120°.  7. 

30°,  210°.  9.  60°,  300°. 

11.  -iVs.  13.  iV2. 

6.  45°,  135°.  8. 

90°,  270°.  10.  135°,  225°. 

12.  |.  14.  |V2. 

19.  60°,  240°, 

22.  19°,  161°, 

25.  19°  28' 17", 

420°,  600°. 

379°,  521°. 

160°  31'  43". 

20.  58°,  238°, 

23.  15°  24'  30",  195°  24'  30", 

26.  ± lV2. 

418°,  598°. 

375°  24'  30",  555°  24'  30" 

27.  ± |VS  or  0. 

21.  74°,  106°, 

24.  19°,  341°, 

434°,  466°. 

379°,  701°. 

Exercise  74.  Page  161 

2.  360°  or  2 tt. 

6.  180°  or  •7T. 

9.  180°  and  360°. 

4.  180°  or  7T. 

8.  360°  or  2 tt. 

10.  Complements. 

Exercise  75.  Page  162 

1.  270.63. 

9.  40' 9". 

13.  ^ radian ; 

2.  416.65. 

10.  - 175°,  185°, 

19°  5'  55". 

3.  2695.8. 

535°,  545°. 

22.  30°,  210°, 

4.  4.163. 

11.  - 200°,  160°, 

390°,  570°. 

5.  Impossible. 

560°,  520°. 

23.  60°,  240°, 

6.  Impossible. 

12.  2 radians ; 

420°,  600°. 

7.  345.48  ft. 

114°  35'  30". 

Exercise  77.  Page  166 


1. 

l-n-or  %TT. 

16. 

26°  34'  or  206°  34'. 

2. 

90°  or  270°. 

17. 

30°  or  150°. 

3. 

21°  28'  or  158°  32'. 

18. 

4.5°  or  135°. 

4. 

0°  or  90°. 

19. 

60°,  90°,  270°,  or  300°. 

5. 

30°,  150°,  199°  28',  or  340°  32'. 

20. 

60°,  90°,  120°,  240°, 

, 270°,  or 

6. 

51°  19',  180°,  or  308°  41'. 

300°. 

7. 

30°,  150°,  or  270°. 

21. 

32°  46',  147°  14',  212°  46', 

,01-327°  14'. 

8. 

35°  16',  144°  44',  215°  16',  or  324°  44 

9. 

75°  58'  or  255°  58'. 

Ldll  • 

2 a 

10. 

60°,  180°,  or  300°. 

23. 

. / — a + Va'^  + 8 a + 8\ 

11. 

90°  or  143°  8'. 

cos--^  

V 4 ' 

)■ 

13. 

30°,  150°,  210°,  or  330°. 

24. 

1. 

13. 

0°,  120°,  180°,  or  240°. 

25. 

1. 

14. 

45°,  161°  34',  225°,  or  341°  34'. 

26. 

0°,  45°,  90°,  180°,  225°, 

or  270°. 

15. 

60°,  120°,  240°,  or  300°. 

27. 

30°,  150°,  210°,  or  330°. 

20 


PLANE  TRIGONOMETRY 


28. 

30°,  60°,  120°,  150°,  210°,  240°, 

CO 

o 

o 

o 

60. 

60°,  90°,  120°,  240°,  270°,  or  300°. 

or  330°. 

61. 

0°,  90°,  180°,  or  270°. 

29. 

0°,  65°  42^  180°,  or  204°  18'. 

63. 

0°,  90°,  120°,  180°,  240°, 

or  270°. 

30. 

14°  29',  30°,  150°,  or  165°  31'. 

63. 

0°,  74°  5',  127°  25',  180 

°,  232°  35', 

31. 

0°,  20°,  100°,  140°,  180°,  220°, 

260°, 

or  285°  55'. 

or  340°. 

64. 

0°,  180°,  220°  39',  or  319°  21'. 

32. 

45°,  90°,  135°,  225°,  270°,  or  315°. 

65. 

8°  or  168°. 

33. 

30°,  150°,  or  270°. 

66. 

40°  12',  1.39°  48',  220°  12', 

or  319°  48'. 

34. 

26°  34',  90°,  206°  34',  or  270°. 

67. 

0°,  60°,  120°,  180°,  240°, 

or  300°. 

35. 

45°,  135°,  225°,  or  315°. 

68’. 

30°  or  330°. 

36. 

45°,  135°,  22.5°,  or  315°. 

69. 

60°,  120°,  240°,  or  300°. 

37. 

15°,  7.5°,  135°,  195°,  255°,  or  31-5°. 

70. 

18°,  90°,  162°,  2.34°,  270°, 

, or  306°. 

38. 

45°,  135°,  225°,  or  315°. 

71. 

30°,  00°,  120°,  150°,  210°, 

, 240°,  300°, 

39. 

0°,  45°,  180°,  or  22-5°. 

or  330°. 

40. 

0°,  90°,  120°,  240°,  or  270°. 

72. 

53°  8',  126°  52',  233°  8', 

or  306°  52'. 

41. 

0°,  36°,  72°,  108°,  144°,  180°, 

216°, 

73. 

30°. 

252°,  288°,  or  324°. 

74. 

22°  37'  or  143°  8'. 

42. 

120°. 

75. 

0°,  20°,  30°,  40°,  60°,  80°,  90°,  100°, 

43. 

54°  44',  125°  16',  234°  44',  .30.5°  16'. 

120°,  140°,  1.50°,  160°, 

180°,  200°, 

44. 

30°,  60°,  90°,  120°,  150°,  210°, 

240°, 

210°,  220°,  240°,  260°, 

270°,  280°, 

270°,  300°,  or  330°. 

300°,  320°,  330°,  or  340°. 

45. 

■ 1 I*  - 1 

76. 

221°,  45°,  671°,  90°,  1121°,  1350^ 

2 ■ 

157i°,  2021°,  22.5°,  247^°,  270°, 

46. 

90°,  216°  52',  or  323°  8'. 

292  i°,  315°,  or  337^' 

47. 

30°,  90°,  150°,  210°,  270°,  or  330°. 

77. 

45°  or  22.5°. 

48. 

0°,  45°,  180°,  or  225°. 

78. 

± 1 or  ± 1 ViT. 

49. 

45°,  60°,  120°,  135°,  225°,  240°, 

, 300°, 

79. 

^ V3  or  — J Vs. 

or  315°. 

80. 

0 or  ± 1. 

50. 

0°,  45°,  135°,  225°,  or  315°. 

81. 

0°,  30°,  90°,  150°,  180°, 

210°,  270°, 

51. 

90°  or  270°. 

or  330°. 

52. 

lV3. 

82. 

120°  or  240°. 

53. 

i. 

83. 

60°,  120°,  240°,  or  300°. 

54. 

6°,  45°,  90°,  180°,  225°,  or  270°. 

84. 

10°  12',  34°  48',  190°  12', 

or  214°  48'. 

55. 

30°,  150°,  210°,  or  330°. 

85. 

29°19',  105°41',  209°19', 

or285°41'. 

56. 

60°. 

86. 

0°,  45°,  90°,  180°,  225°, 

or  270°. 

57. 

105°  or  345°. 

87. 

0°,  45°,  135°,  225°,  or  315°. 

58. 

135°,  315°,  or  -3,  sin-i(l  - a). 

88. 

0°,  60°,  120°,  180°,  240°, 

, or  300°. 

59. 

30°,  60°,  120°,  150°,  210°,  240°, 

, 300°, 

89. 

27°  58',  135°,  242°  2',  or 

315°. 

or  330°. 

Exercise  78. 

Page  170 

1. 

X = a,  y = 0;  or  x = 0,  y = 

a. 

4. 

X = 100,  y = 200. 

in  —71  + 1 

1 la  — b 

5. 

X=:Sin-l±A  

■9 

2. 

x-sin  ^ , 

\ 2 

m + 71  — 1 

■ 1 , /« + ^ 

2 ■ 

y -sin  1 ^ ■ 

6. 

X = 90°, 

3. 

a;  ^ 76°  10',  y = 15°  30'. 

y-0°  or  180°. 

ANSWERS 


21 


1.x  — cos-1  j- V6  — a-  + 2)  ; y = cos-i  l(a  ±V6  — a-  + 2). 

X = tan- tan  a + 1 cos-i  [2  — {2  m-  — 2 n-)  cos- a — 1] ; 

y = tan-i  ^ tan  a — 1 cos-i  [2  m'^  — (2  — 2n^)  cos^a  — 1]. 

9,  a:  = tan-i-  + cos-i  i-  Va^  + 6^ ; ?/  = tan-i  - — cos-i  1 Va^  + 6^. 
b - b " 

10.  2 = 24°  13',  r = 225.12;  2 = 204°  13',  r=-  225.12. 

11.  2 = 42° 28',  r = 151 ; 2 = 222° 28',  r=-  151. 


Exercise  79.  Page  171 

1.  ^ = 30°  or  1-50°;  2 = 0.134  or  1.866. 

2.  B = sin-1  (a  — i) ; x = 2 — a. 


3.  X = 45°,  135°,  225°,  or  315°; 


4.  0 sin-1 
<p=  \ sin- 
b.  6 = eos-i 


a2  + 62 


— 1 1 + i-  sin- 1 


/a2  + 62  . j 

“i| Isin-i 


6.  = 0°. 

Exercise  80. 

1.  a2+  62-2(a-6)  = -l. 

2.  ab  = 1. 

3.  (n  — ?n)2  + (g  — pY  = 1- 

4.  6 — a = i Vp2  g2_ 

P 

5.  6c  = 1. 

6.  2 = ± V2  ry  — 2/^  + r versin-i  - . 


p = 30°,  1-50°,  210°,  or  330°. 
a-  — b- 
a-  + b-  ’ 

«2  - 62 
a2  + 62 

^ a 1 

^ \ 6 - a J 

Page  172 

7.  {m-  + n2  — 1)2  = (n  + 1)2  7^2. 

8.  a-bs  -f-  as63  = l. 

9.  (m.  + n)  V4  — {m  — ?i)2  = 2 (m  — n) 

10.  _p'r  = — r'p. 

11.  k*  + l*  = 2kl{kl-2). 

12.  a^bh'^  + ofic-q-  + 6'ic2p2  = a-W'C^. 


Exercise  81.  Page  176 

1.  1 ; — 1. 

2.  1 ; - 1 ; - V^. 


3.  1;  0.7660  + 0.6428  i;  0.17.36  + 0.9848  i. 

4.  1;  1 ( V5  - 1 + i VlO  + 2 V5  ) ; 1 (- ^/5  - 1 + i ’^lO  - 2 -Vs)  ; 
l(_-v/5-l--iVlO-  2V5);  l(V5-l-iVlO  + 2V5). 


5. 


i + hV^;  -1;  - i- 

1 ; V—  1 ; — ^ 


- i-V^. 

— i VI  — ^ V—  1 ; — V— 1; 


1V3-  -iV-  1. 
6.  ^ V2  + tV  V—  2 ; 


; - ^ V2  - 1-  i V 2 - 1 V^. 


22 


PLANE  TRIGONOMETRY 


Exercise  82.  Page  177 

1.  - 

2.  §V2+  + |V-2;  - fVl  - |V^;  |V2- 

f + f ~ f + I 3 ; — 3. 

4.  2 (cos  36°  + i sin  36°) ; 2 (cos  72°  + i sin  72°) ; 2 (cos  108°  + i sin  108°). 

5.  0.9980  + 0.06281;  0.9921  + 0.12531;  0.9823  + 0.18741. 


Exercise  83.  Page  183 


7. 

120. 

18. 

1.64871. 

28. 

tan  56°  40'  12' 

8. 

5040. 

19. 

cos  28°  39'. 

29. 

tan  28°  38'  20' 

9. 

720. 

20. 

cos  7°  10'. 

30. 

tan  86°  23'  16' 

10. 

40,320. 

21. 

cos  114°  2-5'  32". 

35. 

0.6931  + 2 TTi 

; 0.6931  + Am. 

11. 

3,628,800. 

22. 

cos  0°. 

36. 

1.3862  + 2 TTt 

; 1.3862  + 4 TTi. 

12. 

604,800. 

23. 

sin  57°  17'  48". 

37. 

0.3465  + 2 TTi 

; 0.3465  + 4 771. 

13. 

90. 

24. 

sin  28°  38'  40". 

88. 

0.6931  + TTI ; 

0.6931  + 3 m. 

14. 

42. 

25. 

sin  05°  24'  45"  or 

39. 

1.009  + 2 TTi ; 

1.609  + 4 TTi ; 

15. 

15. 

sin  114°  35' 15". 

1.609  + 6m. 

16. 

6840. 

26. 

sin  0°  or  sin  180°. 

40. 

3.218  + 2 7ri; 

3.218  + 477i; 

17. 

7.38883. 

27. 

tan  0°. 

3.218  + 6 m. 

41. 

4.827  + 2 7Ti  ; 4.827  + 

4:iri ; i.- 

827  + 6 m. 

42. 

1.009  + 7t1;  1.609  + 3 

TTt  ; 1.609  + 5 TTi. 

43. 

4.605170  + 2 ttI  ; 4.605170  + 4 

7TI. 

44. 

2.302585  + TTi ; 2.302585  + 3 iri. 

45. 

6.907755  + 2 7rl;  6.907 

'755  + 4 

ttI, 

46. 

1.151292  + 2 7t1;  1.151292  + 4 

7tL 

Exercise  84. 

Page  184 

1. 

362.8  ft. 

2m  {rfi 

-L  + 2 

n (m-  — 1) 

12.  6 sin  C. 

2.  1445.6/ ft. ; 1704.7  ft.;  ’ (nr  — 1)  (/i- — 1)  — 4mn  ' 13-  794.73  ft. 

1622.5  ft.  5.  2. 


TRIGONOMETRIC  AND 
LOGARITHMIC  TABLES 


BY 

GEORGE  WENTWORTH 
and 

DAVID  EUGENE  SMITH 


GINN  AND  COMPANY 


BOSTON  • NEW  YORK  • CHICAGO  • LONDON 
ATLANTA  • DALLAS  • COLUMBUS  • SAN  FRANCISCO 


COPYRIGHT,  1914,  BY  GEORGE  WENTWORTH 
AND  DAVID  EUGENE  SMITH 
ALL  RIGHTS  RESERVED 

PRINTED  IN  THE  UNITED  STATES  OF  AMERICA 

526.1 


tCbe  gtftenttam 


GINN  AND  COMPANY*  PRO- 
PRIETORS • BOSTON  • U.S.A. 


PEEFACE 


In  preparing  this  new  set  of  tables  for  the  use  of  students  of 
trigonometry  care  has  been  taken  to  meet  the  modern  requirements 
in  every  respect,  while  preserving  the  best  features  to  be  found  in 
those  tables  that  have  stood  the  test  of  long  use.  In  our  country 
the  large  majority  of  teachers  prefer  five-place  logarithmic  tables, 
and  for  this  preference  they  have  cogent  reasons.  While  a five-place 
table  gives  the  results  to  a degree  of  approximation  closer  than  is 
ordinarily  required,  nevertheless  if  a student  can  use  such  a table  it 
is  a simple  matter  to  use  one  with  four  or  six  places.  One  who  has 
been  brought  up  to  use  a table  with  only  four  places,  however,  finds 
it  less  easy  to  adapt  himself  to  a table  having  a larger  number  of 
places.  On  this  account  the  basal  tables  of  logarithms  given  in  this 
book  have  five  decimal  places.  For  the  natural  functions,  however, 
four  decimal  places  are  quite  sufficient  for  the  kind  of  applications 
that  the  student  will  meet  in  his  work  in  trigonometry,  and  the  gen- 
eral custom  of  using  four  places  has  been  followed  in  this  respect. 

Following  the  usage  found  in  the  best  tables,  unnecessary  figures 
have  been  omitted,  thus  relieving  the  eye  strain.  Where,  as  on 
page  28,  the  first  two  figures  of  a mantissa  are  the  same  for  several 
logarithms,  these  figures  are  given  only  in  the  line  in  which  they 
first  occur  and  in  the  lines  corresponding  to  multiples  of  five.  Where, 
however,  a table  is  to  be  read  from  the  foot  of  the  page  upwards,  as 
well  as  from  the  top  downwards,  the  first  two  figures  are  given  both 
at  the  bottom  and  at  the  top  of  the  vacant  space,  as  on  page  51,  so 
that  the  computer  may  have  no  difficulty  in  seeing  them  in  what- 
ever direction  the  eye  is  moving  over  the  table. 

It  will  also  be  seen  that  great  care  has  been  bestowed  upon  the 
selection  of  a type  that  will  relieve  the  eye  from  fatigue  as  far  as 
possible,  and  upon  an  arrangement  of  figures  that  will  assist  the 
computer  in  every  way.  It  is  believed  that  this  care,  together  with 
the  attention  given  to  spacing  and  to  the  general  appearance  of 
the  page,  has  resulted  in  the  most  usable  set  of  trigonometric  and 
logarithmic  tables  that  has  thus  far  been  printed. 

iii 


IV 


PREFACE 


In  recognition  of  the  tendency  at  the  present  time  to  use  four-place 
tables  in  certain  lines  of  work,  Table  I has  been  prepared.  Teachers 
are  advised,  however,  for  the  reasons  already  stated,  to  use  the  five- 
place  table  first  and  until  it  is  clearly  understood,  taking  Table  I 
for  the  work  that  requires  only  a low  degree  of  approximation. 

The  tendency  to  use  decimal  parts  of  a degree  instead  of  minutes 
and  seconds  is  one  that  will  undoubtedly  increase.  This  tendency 
is  therefore  recognized  by  the  introduction  of  a conversion  table. 
By  its  use  the  student  can  instantly  adapt  the  common  tables  to 
the  decimal  plan.  At  the  same  time  it  is  apparent  that  students 
will  be  called  upon  to  use  the  sexagesimal  division  of  the  degree 
almost  exclusively  for  years  to  come,  and  for  this  reason  the  empha- 
sis should  be  placed,  as  it  is  in  the  authors’  Plane  and  Spherical 
Trigonometry,  upon  the  sexagesimal  instead  of  the  decimal  division. 

It  is  confidently  believed  that  teachers  and  students  will  find  in 
these  tables  all  that  they  need  for  the  purposes  of  the  computation 
required  in  every  line  of  work  in  trigonometry. 


GEOKGE  WENTWORTH 
DAVID  EUGENE  SMITH 


CONTENTS 


PAGE 

Lnteoduction 1 

Table  I.  Four-Place  Mantissas  of  Logarithms  of 

Integers  and  Trigonobietric  Functions  . 17 

Table  II.  Circumferences  and  Areas  of  Circles  . . 24 

Table  III.  Five-Place  Mantissas  of  Logarithms  of 

Integers  from  1 to  10,000  27 

Table  IV.  Proportional  Parts 46 

Table  V.  Logarithms  of  Constants . 48 

Table  VI.  Logarithms  of  Trigonometric  Functions  . 49 

Table  VII.  Corrections  for  Small  Angles 78 

Table  VIII.  Natural  Functions 79 

Table  IX.  Conversion  of  Degrees  to  Eadians  . . . 102 

Table  X.  Conversion  of  Minutes  and  Seconds  to 
Decimals  of  a Degree,  and  of  Decimals 
OF  A Degree  to  Minutes  and  Seconds  . . 104 


7 


INTRODUCTION 


1.  Logarithm.  The  power  to  which  a given  number,  called  the 
base,  must  be  raised  to  equal  another  given  number  is  called  the 
logarithm  of  this  second  given  number. 

For  example,  since  10^  = 1000, 

therefore,  to  the  base  10,  3 = the  logarithm  of  1000. 

In  this  case  1000  is  called  the  antilogarithm  of  3,  this  being  the  number 
corresponding  to  the  logarithm. 

In  this  Introduction  only  the  most  important  facts  relating  to  logarithms 
are  given.  For  a more  complete  treatment  see  the  Wentworth-Smith  Plane 
and  Spherical  Trigonometry,  Chapter  III. 


2.  Symbolism.  For  " logarithm  of  N ” it  is  customary  to  write 
log  N,  If  we  wish  to  specify  log  N to  the  base  b we  write  logj  N, 
reading  this  " logarithm  of  N to  the  base  b.” 

For  example,  since  2®  = 8,  we  see  that  logjS  = 3 ; and  since  6^  = 25,  logj25  = 2. 


3.  Base.  We  may  take  various  bases  for  systems  of  logarithms, 
but  for  practical  calculation  in  trigonometry,  10  is  taken  as  the  base. 

Logarithms  are  due  chiefly  to  John  Napier  of  Scotland  (1614),  but  the 
base  10  was  suggested  by  Henry  Briggs  of  Oxford.  Hence  logarithms  to  the 
base  10  are  often  called  Briggs  logarithms. 


4.  Logarithm  of  a Product.  The  logarithm  of  the  'product  of  several 
numbers  is  equal  to  the  sum  of  the  logarithms  of  the  numbers. 


For  if 
and  if 

Therefore 
For  example, 


H = 10^,  then  x = logH; 

B = lO^',  then  y = logB. 
AB  = \Qr^+y,  and  x + y — \ogAB. 
log  (247  X 7.21)  = log  247  + log  7.21. 


5.  Logarithm  of  a Quotient.  The  logarithm  of  the  quotient  of  t-wo 
numbers  is  equal  to  the  logarithm  of  the  dividend  minus  the  logarithm 
of  the  divisor. 

For  if  A = l(y^,  then  x = log.d.; 

and  if  B=  10^,  then  y = log  B. 

A A 

Therefore  — and  x — y = log  — • 

B ^ °B 

log  (9.2  -H  6.7)  = log  9.2  - log  6.7. 

1 


For  example, 


2 


TABLES 


6.  Logarithm  of  a Power.  The  logarithm  of  a power  of  a number 
is  equal  to  the  logarithm  of  the  number  multiplied  by  the  exponent. 
For  if  x = logA,  then  A — 1(F. 

Raising  to  the  pth  power,  Ap  = IQp^. 

Hence  log  Ap  = px  = plogA. 

For  example,  log  7.2^  = 5 log  7.2. 


7.  Logarithm  of  a Root.  The  logar  ithm  of  a root  of  a number  is  equal 
to  the  logarithm  of  the  number  divided  by  the  index  of  the  root. 

For  if  X = logH,  then  H = ICF. 


Taking  the  rth  root. 
Hence 

For  example. 


A’-  = 10''. 


X logH 


logH''  = - = 

r , 

log ^ log  9.36. 


8.  Characteristic  and  Mantissa.  Usually  a logarithm  consists  of  an 
integer  plus  a decimal  fraction. 

The  integral  part  of  a logarithm  is  called  the  characteristic. 

The  decimal  part  of  a logarithm  is  called  the  mantissa. 

Thus,  if  log  2353  = 3.37162,  the  characteristic  is  3 and  the  mantissa  is 
0.37162.  This  means  that  lOS-s’i®^  = 2353,  or  that  the  100,000th  root  of  the 
337,162d  power  of  10  is  approximately  2353. 

The  logarithms  of  integral  powers  of  10  are,  of  course,  integers,  the  mantissa 
in  every  such  case  being  zero.  For  example,  since  1000  = 10^,  log  1000  = 3. 


9.  Finding  the  Characteristic.  The  characteristic  is  not  usually 
given  in  a table  of  logarithms,  because  it  is  easily  found  mentally. 

The  characteristic  of  the  logarithm  of  a number  greater  than  1 is 
positive  and  is  one  less  than  the  number  of  integral  places  in  the 
number. 

The  characteristic  of  the  logarithm  of  a number  between  0 and  1 is 
negative  and  is  one  greater  than  the  number  of  zeros  between  the  deci- 
mal point  and  the  first  significant  figure  in  the  number. 

For  example,  since  10®  = 1000  and  10^  = 10,000,  it  is  evident  that  log  7250 
lies  between  3 and  4. 

For  further  explanation  see  the  Wentwofth-Smith  Plane  Trigonometry,  § 46. 


10.  The  Negative  Characteristic.  The  mantissa  is  always  consid- 
ered as  positive.  If  log  0.02  = — 2 -t-  0.30103,  we  cannot  write  it 
— 2.30103  because  this  would  mean  that  both  mantissa  and  character- 
istic are  negative.  Hence  the  form  2.30103  has  been  chosen,  which 
means  that  only  the  characteristic  2 is  negative. 

In  practical  computation  it  is  more  often  written  0.30103  — 2,  or  8.30103—10, 
but  when  written  by  itself  the  form  2.30103  is  convenient. 


INTRODUCTION 


3 


11.  Mantissa  independent  of  Decimal  Point.  The  mantissa  of  the 
logarithm  of  a number  is  unchanged  by  any  change  in  the  position 
of  the  decimal  point  of  the  number. 

Tor  if  103.37107  = 23  50,  then  log  2350  = 3.37107. 

Dividing  by  10,  102-3™7  = 235,  and  log  235  = 2.37107. 

That  is,  the  mantissa  of  log  2350  is  the  same  as  that  of  log  235.0,  and  so  on, 
wherever  the  decimal  point  is  placed. 

This  is  of  great  importance,  for  if  the  table  gives  the  mantissa  for  only  235, 
we  know  that  this  is  the  mantissa  for  0.235,  2.35,  23.5,  235,000,  and  so  on. 

12.  Logarithms  Approximate.  Logarithms  are,  in  general,  only 
approximate.  Although  log  1000  is  exactly  3,  log  7 is  only  approxi- 
mately 0.84510. 

To  four  decimal  places,  log  7 = 0.8451 ; to  five  decimal  places,  0.84510 ; to 
six  decimal  places,  0.845098,  and  so  on. 

In  a four-place  table  there  is  a possible  error  of  ^ of  0.0001 ; in  a five-place 
table,  of  ^ of  0.00001,  and  so  on,  but  in  each  case  the  probable  error  is  much  less. 

If  several  logarithms  are  added  the  possible  error  is  correspondingly  increased. 

In  finding  antilogarithms  the  first  figure  found  by  interpolation  is  usually 
accurate,  the  second  is  doubtful,  and  the  third  is  rarely  trustworthy. 

13.  Cologarithm.  The  logarithm  of  the  reciprocal  of  a number  is 
called  the  cologarithm  of  the  number. 

The  cologarithm  of  x is  expressed  thus  ; colog  x. 

Since  colog  x = log  - = log  1 — log  x = 0 — logx,  we  have 

X 

colog  X = — log  X. 

That  is,  colog  2 = — log  2. 

To  avoid  a negative  mantissa  this  may  be  written 

colog  X = 10  — log  X — 10. 

For  example,  colog  2 = — log  2 = 10  — 0.30103  — 10 

= 9.69897  - 10. 

14.  Use  of  the  Cologarithm.  Since  to  divide  by  a number  is  the 
same  as  to  multiply  by  its  reciprocal,  instead  of  subtracting  the 
logarithm  of  a divisor  we  may  add  its  cologarithm. 

The  cologarithm  of  a number  is  easily  written  by  looking  at  the  logarithm 
in  the  table.  Thus,  since  log  20  = 1.30103,  we  find  colog  20  by  mentally  sub- 
tracting this  from  10.00000  — 10.  This  is  done  by  beginning  at  the  left  and 
subtracting  the  number  represented  by  each  figure  from  9,  except  the  right- 
hand  figure,  which  we  subtract  from  10. 

For  example,  if  we  have  to  simplify 

625  X 7.51 
2.73  X 14  8’ 

it  is  easier  to  add  log  625,  log  7.51,  colog  2.73,  and  colog  14.8,  than  to  add  log 
625  and  log  7.51,  and  then  to  add  log  2.73  and  log  14.8,  and  finally  to  subtract. 


4 


TABLES 


15.  General  Use  of  the  Tables.  In  writing  down  a logarithm  always 
write  the  characteristic  before  looking  for  the  mantissa.  Otherwise 
the  characteristic  may  be  forgotten. 

Some  computers  find  it  convenient  to  paste  paper  tabs  so  that 
they  project  from  the  side  of  the  first  page  of  each  table,  thus  allow- 
ing the  book  to  be  opened  quickly  at  the  desired  table. 

Although  a table  of  proportional  parts  is  given,  it  is  best  to  ac- 
custom the  eye  to  interpolate  quickly  from  the  regular  table. 

TABLE  I 

16.  Nature  of  Table  I.  This  is  a table  of  logarithms  of  integers 
from  1 to  1000,  and  of  the  sine,  cosine,  tangent,  and  cotangent,  the 
mantissas  extending  to  four  decimal  places  and  the  characteristics 
being  10  too  large,  as  in  Table  VI.  For  the  ordinary  computations 
of  physics  and  mensuration  this  is  sufficient,  the  results  in  general 
being  correct  to  four  figures. 

There  is  a growing  disposition  to  use  the  convenient  four-place  table  for 
ordinary  work.  Most  teachers  prefer,  however,  to  use  a five-place  table,  since 
the  student  who  can  use  this  will  have  no  trouble  with  the  simpler  four-place 
table.  For  this  reason  the  computations  in  the  Wentworth-Smith  Plane  and 
Spherical  Trigonometry  are  based  upon  the  five-place  table. 

17.  Arrangement  of  the  Table.  The  vertical  columns  headed  N con- 
tain the  numbers,  and  the  other  columns  the  logarithms.  On  page  17 
the  characteristics  as  well  as  the  mantissas  are  given,  but  on  pages  18 
and  19  only  the  mantissas  are  given,  the  characteristics  being  deter- 
mined by  § 9.  To  find  the  mantissa  for  16,  look  on  the  line  to  the  right 
of  16  and  in  the  column  marked  O.  This  mantissa,  0.2041,  is  the  same 
as  that  for  1.6, 160, 1600,  and  so  on.  To  find  the  mantissa  for  167,  look 
on  the  line  to  the  right  of  16  and  in  the  column  marked  7.  This  man- 
tissa, 0.2227,  is  the  same  as  that  for  0.167,  16.7,  167,000,  and  so  on. 

The  table  of  trigonometric  functions  is  arranged  for  every  10',  this 
being  sufiicient  for  many  practical  purposes. 

18.  To  find  a Logarithm  or  Antilogarithm.  The  method  of  finding 
the  logarithm  of  a number  or  the  antilogarithm  of  a logarithm  is 
the  same  as  that  employed  with  a five-place  table  (§§  21-24). 

TABLE  II 

19.  Nature  of  Table  II.  This  table  (pages  24  and  25)  contains  the 
circumferences  and  areas  of  circles  of  given  radii,  and  the  diam- 
eters of  circles  of  given  circumference  or  given  area.  It  often  saves 
a considerable  amount  of  computation  in  problems  involving  circles, 
cylinders,  spheres,  and  cones. 


ENTRODUCTION 


5 


TABLE  III 

20.  Arrangement  of  Table  III.  In  this  table  (pages  27-45)  the  ver- 
tical columns  headed  N contain  the  numbers,  and  the  other  columns 
the  logarithms.  On  page  27  both  the  characteristic  and  the  mantissa 
are  printed.  On  pages  28-45  the  mantissa  only  is  printed,  and  the 
decimal  point  and  unnecessary  figures  are  omitted  so  as  to  relieve 
the  eye  from  strain. 

The  fractional  part  of  a logarithm  is  only  approximate,  and  in  a 
five-place  table  all  figures  that  follow  the  fifth  are  rejected. 

Thus,  if  the  mantissa  of  a logarithm  written  to  seven  places  is  5326143  it  is 
written  in  this  table  (a  five-place  table)  53261.  If  it  is  5329788  it  is  written 
53298.  If  it  is  5328461  or  5328499  it  is  written  in  this  table  53285.  If  the  man- 
tissa is  5325506  it  is  written  53255  ; and  if  it  is  5324486  it  is  written  53245. 

21.  To  find  the  Logarithm  of  a Number.  If  the  given  number  con- 
sists of  one  or  two  significant  figures,  the  logarithm  is  given  on 
page  27.  If  zeros  follow  the  significant  figures,  or  if  the  number 
is  a proper  decimal  fraction,  the  characteristic  must  be  determined. 

If  the  given  number  has  three  significant  figures,  it  will  be  found 
in  the  column  headed  N (pages  28-45)  and  the  mantissa  of  its  loga- 
rithm will  be  found  in  the  next  column  to  the  right. 

For  example,  on  page  28,  log  145  = 2.16137,  and  log  14500  = 4.16137. 

If  the  given  number  has  four  significant  figures,  the  first  three 
will  be  found  in  the  column  headed  N,  and  the  fourth  will  he  found 
at  the  top  of  the  page  in  the  line  containing  the  figures  1,  2,  3,  etc. 
The  mantissa  will  be  found  in  the  column  headed  by  the  fourth  figure. 

For  example,  on  pages  41  and  44  we  find  the  following  : 
log  7682  = 3.88547,  log  76.85  = 1.88564  ; 

log 93280  = 4.96979,  log  0.9468  = 9.97626  - 10. 

22.  Interpolation  for  Logarithms.  If  the  given  number  has  five 
or  more  significant  figures,  a process  called  interpolation  is  required. 

Interpolation  is  based  on  the  assumption  that  between  two  consecutive  man- 
tissas of  the  table  the  change  in  the  mantissa  is  directly  proportional  to  the 
change  in  the  number.  This  assumption  is  not  exact,  but  the  error  does  not, 
in  general,  affect  the  first  figure  found  in  this  manner. 

For  example,  required  the  logarithm  of  34237. 

The  required  mantissa  is  (§  11)  the  same  as  the  mantissa  for  3423.7 ; therefore 
it  will  be  found  by  adding  to  the  mantissa  of  3423  seven  tenths  of  the  difference 
between  the  mantissas  for  3423  and  3424. 

The  mantissa  for  3423  is  53441,  and  the  mantissa  for  3424  is  53453. 

The  difference  between  these  mantissas  (tabular  difference)  is  12. 

Hence  the  mantissa  for  3423.7  is  53441  -b  (0.7  of  12)  = 53449. 

Therefore  the  required  logarithm  of  34237  is  4.53449. 


6 


TABLES 


23.  To  find  the  Antilogarithm.  If  the  given  mantissa  can  be  found 
in  the  table,  the  first  three  significant  figures  of  the  required  number 
will  be  found  in  the  column  headed  N in  the  same  line  with  the 
mantissa,  and  the  fourth  figure  at  the  top  of  the  column  containing 
the  mantissa.  The  position  of  the  decimal  point  is  determined  by 
the  characteristic  (§9). 

1.  Find  the  antilogarithm  of  0.92002. 

The  number  for  the  mantissa  92002  is  8318.  (Page  42.) 

The  characteristic  is  0 ; therefore  the  required  number  is  8.318. 

2.  Find  the  antilogarithm  of  6.09167. 

The  number  for  the  mantissa  09167  is  1235.  (Page  28.) 

The  characteristic  is  6 ; therefore  the  required  number  is  1,235,000. 

3.  Find  the  antilogarithm  of  7.50325  — 10. 

The  number  for  the  mantissa  50325  is  3186.  (Page  32.) 

The  characteristic  is  — 3 ; therefore  the  required  number  is  0.003186. 

24.  Interpolation  for  Antilogarithms.  If  the  given  mantissa  cannot 
be  found  in  the  table,  find  in  the  table  the  two  adjacent  mantissas 
between  which  the  given  mantissa  lies,  and  the  four  figures  corre- 
sponding to  the  smaller  of  these  two  mantissas  will  be  the  first  four- 
significant  figures  of  the  required  number.  If  more  than  four  figures 
are  desired,  they  may  be  found  by  interpolation,  as  in  the  following 
examples : 

1.  Find  the  antilogarithm  of  1.48762. 

Here  the  two  adjacent  mantissas  of  the  table,  between  which  the  given  man- 
tissa 48762  lies,  are  found  to  be  (page  32)  48756  and  48770.  The  antilogarithms 
are  3073  and  3074.  Tne  smaller  of  these,  3073,  contains  the  first  four  significant 
figures  of  the  required  number. 

The  difference  between  the  two  adjacent  mantissas  is  14,  and  the  diSerence 
between  the  corresponding  numbers  is  1. 

The  difference  between  the  smaller  of  the  two  adjacent  mantissas,  48756,  and 
the  given  mantissa,  48762,  is  6.  Therefore  the  number  to  be  annexed  to  3073 
is  ^ of  1,  which  is  0.43,  and  the  fifth  significant  figure  of  the  required  anti- 
logarithm  is  4. 

Hence  the  required  antilogarithm  is  30.734. 

2.  Find  the  antilogaritbm  of  7.82326  — 10. 

The  two  adjacent  mantissas  between  which  82326  lies  are  (page  39)  82321 
and  82328.  The  antilogarithm  having  the  mantissa  82321  is  6656. 

The  difference  between  the  two  adjacent  mantissas  is  7,  and  the  difference 
between  the  corresponding  numbers  is  1 . 

The  difference  between  the  smaller  mantissa,  82321,  and  the  given  mantissa, 
82326,  is  5.  Therefore  the  number  to  be  annexed  to  6656  is  4 of  1,  which  is 
0.7,  and  the  fifth  significant  figure  of  the  required  antilogarithm  is  7. 

Hence  the  required  antilogarithm  is  0.0066567. 


INTRODUCTION 


7 


TABLE  IV 

25.  Proportional  Parts.  In  interpolating  (§§  22,  24)  we  have  to 
find  fractional  parts  of  the  difference  between  two  numbers  or  two 
logarithms. 

For  example,  in  finding  log  73.537  we  see  that 

log  73.54  = 1.86652 
log  73.53  = 1.86646 
Tabular  difference  = 6 

tabular  difference  = 4 

Adding  1.86646  and  0.00004,  we  have 

log  73.537  = 1.86650 

These  fractional  parts  of  a tabular  difference  are  called  propor- 
tional parts. 

26.  Nature  of  Table  IV.  In  Table  IV  the  proportional  parts  of  all 
differences  from  1 to  100  are  given,  so  that  by  turning  to  the  table 
we  can  make  any  ordinary  interpolation  at  a glance. 

For  example,  if  the  difference  (D)  is  6,  as  in  the  first  case  considered  in  § 24, 
the  table  shows  that  of  this  difference  is  4.2,  the  last  figure  being  rejected 
because  it  is  less  than  5.  In  such  a simple  case,  however,  we  would  make  the 
interpolation  mentally,  without  reference  to  the  table. 

If  the  difference  were  87,  and  we  wished  of  this  difference,  the  table 
shows  at  once  that  this  is  78.3,  from  which  we  would  reject  the  last  figure 
as  before. 

In  some  sets  of  tables  the  proportional  parts  are  printed  beside  the  loga- 
rithms themselves,  but  this  necessitates  the  use  of  a small  type  that  is  trying 
to  the  eyes.  It  is  usually  easier  to  make  the  interpolation  mentally  than  to  use 
the  table  of  proportional  parts,  but  where  a large  number  of  interpolations  are 
to  be  made  at  the  same  time  the  table  is  helpful. 

27.  Table  IV  for  Multiplication.  By  ignoring  tbe  decimal  points 
Table  IV  may  be  used  as  a multiplication  table,  tbe  column  marked 
D containing  the  multiplicands,  the  multipliers  1-9  appearing  at 
the  top,  and  the  products  being  given  below. 

For  example,  8 x 79  = 632,  as  is  seen  by  looking  to  the  right  of  79  and 
under  8. 


TABLE  V 

28.  Logarithms  of  Constants.  There  are  certain  constants,  such  as 
7T,  2 7T,  V2,  and  so  on,  that  enter  frequently  into  the  computations 

of  trigonometry.  To  save  the  trouble  of  looking  for  the  logarithms 
of  these  numbers  in  the  regular  table,  or  of  computing  their  loga- 
rithms, Table  V has  been  prepared. 


8 


TABLES 


TABLE  VI 

29.  Nature  of  Table  VI.  This  table  (pages  49-77)  contains  the 
logarithms  of  the  trigonometric  functions  of  angles.  In  order  to 
avoid  negative  characteristics,  the  characteristic  of  every  logarithm 
is  printed  10  too  large.  Therefore  — 10  is  to  be  annexed  to  each 
logarithm. 

On  pages  49-55  the  characteristic  remains  the  same  throughout  each  column 
and  is  printed  at  the  top  and  the  bottom  of  the  column ; but  on  pages  56-77 
when  the  characteristic  changes  one  unit  in  value  the  place  of  each  change  is 
marked  with  a bar.  Above  each  bar  the  proper  characteristic  is  printed  at  the 
top  of  the  column  ; below  each  bar  the  characteristic  is  printed  at  the  bottom. 

On  pages  56-77  the  log  sin,  log  cos,  log  tan,  and  log  cot  are  given 
for  every  minute  from  1°  to  89°.  Conversely,  this  part  of  the  table 
gives  the  value  of  the  angle  to  the  nearest  minute  when  log  sin, 
log  cos,  log  tan,  or  log  cot  is  known,  provided  log  sin  or  log  cos  lies 
between  8.24186  and  9.99993,  and  log  tan  or  log  cot  lies  between 
8.24192  and  11.76808. 

If  the  exact  value  of  the  given  logarithm  of  a function  is  not  found  in  the 
table,  the  value  nearest  to  it  is  to  be  taken  unless  interpolation  is  employed 
as  explained  in  § 30. 

If  the  angle  is  less  than  45°  the  number  of  degrees  is  printed  at 
the  top  of  the  page,  and  the  number  of  minutes  in  the  column  to 
the  left  of  the  columns  containing  the  logarithms.  If  the  angle  is 
greater  than  45°  the  number  of  degrees  is  printed  at  the  bottom 
of  the  page,  and  the  number  of  minutes  in  the  column  to  the  right 
of  the  columns  containing  the  logarithms. 

If  the  angle  is  less  than  45°  the  names  of  its  functions  are  printed  at  the 
top  of  the  page  ; if  greater  than  45°,  at  the  bottom  of  the  page.  Thus, 


log  sin  21°  37'  = 9.56631  - 10.  Page  66 

log  cot  36°  53'  = 10.12473  - 10  = 0.12473.  Page  73 

log  cos  69°  14' = 9.54969  — 10.  Page  65 

log  tan  45°  59'  = 10.01491  - 10  = 0.01491.  Page  77 

log  tan  75°  12' = 10.57805 - 10.  Page  62 

log  cos  82°  17'  = 9.12799  - 10.  Page  59 

If  log  cos  X = 9.87468  - 10,  x = 41°  28'.  Page  76 

If  log  cot  X = 9.39353  - 10,  x = 76°  6'.  Page  62 

If  log  sin  X = 9.99579  — 10,  x = 82°  2'.  Page  59 

If  log  tan  X = 9.02162  — 10,  x = 6°.  Page  58 


If  log  sin  = 9.47760  — 10,  the  nearest  log  sin  in  the  table  is  9.47774  — 10 
(page  64),  and  the  angle  corresponding  to  this  value  is  17°  29'. 

If  log  tan  = 0.76520  = 10.76520  — 10,  the  nearest  log  tan  in  the  table  is 
10.76490  — 10  (page  60),  and  the  angle  corresponding  to  this  value  is  80°  15'. 
For  the  method  of  interpolating,  see  § 30. 


LNTEODUCTION 


9 


30.  Interpolation.  If  it  is  desired  to  obtain  the  logaritlim  of  the 
function  of  an  angle  that  contains  seconds,  or  to  obtain  the  value  of 
an  angle  in  degrees,  minutes,  and  seconds  from  a logarithm  of  a func- 
tion, interpolation  must  be  employed.  The  theory  of  interpolation 
has  already  been  given  in  §§22  and  24. 

Here  it  must  be  remembered  that  the  difference  between  two  consecutive 
angles  in  the  table  is  1',  and  that  therefore  a proportional  part  of  60"  must  be 
taken.  It  must  also  be  remembered  that  log  sin  and  log  tan  increase  as  the 
angle  increases,  but  log  cos  and  log  cot  diminish  as  the  angle  increases. 

1.  Find  log  tan  70°  46' 8". 

Log  tan  70°  46'  = 0.45731.  (Page  66.) 

The  difference  between  the  mantissas  of  log  tan  70°  46'  and  log  tan  70°  47' 
is  41,  and  of  41  = 5. 

As  the  function  is  increasing,  the  5 must  be  added  to  the  figure  in  the  fifth 
place  of  the  mantissa  45731  ; therefore  log  tan  70°  46'  8"  = 0.45736. 

2.  Find  log  cos  47°  35'  4". 

Log  cos  47°  35'  = 9.82899  - 10.  (Page  76.) 

The  difference  between  this  mantissa  and  the  mantissa  of  log  cos  47°  36' 
is  14,  and  of  14  = 1. 

As  the  function  is  decreasing,  the  1 must  be  subtracted  from  the  figure  in 
the  fifth  place  of  the  mantissa  82899  ; therefore  log  cos  47°  35'  4"  = 9.82898  — 10. 

3.  Find  X when  log  sin  x = 9.46359  — 10. 

The  mantissa  of  the  nearest  smaller  log  sin  in  the  table  is  45334.  (Page  63.) 

The  angle  corresponding  to  this  value  is  16°  30'. 

The  difference  between  45334  and  the  given  mantissa,  45359,  is  25. 

The  difference  between  45334  and  the  next  following  mantissa,  45377,  is  43 
(the  tabular  difference)  and  of  60"=  35". 

As  the  function  is  increasing,  the  35"  must  be  added  to  16°  30' ; therefore 
the  required  angle  is  16°  30'  35". 

4.  Find  x when  log  cot  x — 0.73478. 

The  mantissa  of  the  nearest  smaller  log  cot  in  the  table  is  73415.  (Page  60.) 

The  angle  corresponding  to  this  value  is  10°  27'. 

The  difference  between  73415  and  the  given  mantissa  is  63. 

The  difference  between  73415  and  the  next  larger  mantissa  is  71  (the 
tabular  difference)  and  of  60"=  53". 

As  the  function  is  decreasing,  the  53"  must  be  subtracted  from  10°  27'; 
therefore  the  required  angle  is  10°  26'  7". 

5.  Find  X wben  log  cos  x = 0.83584. 

The  mantissa  of  the  nearest  smaller  log  cos  in  the  table  is  83446.  (Page  67.) 

The  angle  corresponding  to  this  value  is  86°  5'. 

The  difference  between  83446  and  the  given  mantissa  is  138. 

The  tabular  difference  is  184,  and  of  60"  is  45". 

As  the  function  is  decreasing,  45  " must  be  subtracted  from  86°  6' ; therefore 
X = 86°  5'  - 46",  or  86°  4'  16". 


10 


TABLES 


31.  The  Secant  and  Cosecant.  In  working  with  logarithms  we  very 
rarely  use  either  the  secant  or  the  cosecant ; for  sec  x = 1 /cos  x,  and 
log  sec  X = colog  cos  x.  If,  however,  log  sec  or  log  esc  of  an  angle 
is  desired,  it  may  be  found  from  the  table  by  the  formulas, 

sec  A = — — , hence  log  sec  A = colog  cos  A : 
cos  A 

CSC  A = — ^ — , hence  log  esc  A = colog  sin  A. 
sin  A 

For  example, 

log  sec  8°  28'  = colog  cos  8°  28'  = 0.00476.  Page  59 

log  CSC  18°  36'  = colog  sin  18°  36'  = 0.49626.  Page  64 

log  sec  62°  27'  = colog  cos  62°  27'  = 0.33487.  Page  69 

log  CSC  59°  36'  44"  = colog  sin  59°  36'  44"  = 0.06418.  Page  70 

32.  Functions  of  Small  Angles.  If  a given  angle  is  between  0°  and 
1°,  or  between  89°  and  90° ; or,  conversely,  if  a given  log  sin  or 
log  cos  does  not  lie  between  the  limits  8.24186  and  9.99993  in  the 
table ; or  if  a given  log  tan  or  log  cot  does  not  lie  between  the 
limits  8.24192  and  11.75808  in  the  table,  — then  pages  49-55  of 
Table  VI  must  be  used. 

On  page  49,  log  sin  of  angles  between  0°  and  0°  3’,  and  log  cos  of 
the  complementary  angles  between  89°  57'  and  90°,  are  given  to 
every  second ; for  the  angles  between  0°  and  0°  3',  log  tan  = log  sin, 
and  log  cos  = 0.00000 ; for  the  angles  between  89°  57'  and  90°, 
log  cot  = log  cos,  and  log  sin  = 0.00000. 

On  pages  50-52,  log  sin,  log  tan,  and  log  cos  of  angles  between 
0°  and  1°,  or  log  cos,  log  cot,  and  log  sin  of  the  complementary 
angles  between  89°  and  90°,  are  given  to  every  10". 

When  log  tan  and  log  cot  are  not  given,  they  may  he  found  by  the  formulas, 
log  tan  = colog  cot.  log  cot  = colog  tan. 

Conversely,  if  a given  log  tan  or  log  cot  is  not  contained  in  the  table,  then 
the  colog  must  be  found  ; this  will  be  the  log  cot  or  log  tan,  as  the  case  may  be, 
and  will  be  contained  in  the  table. 

On  pages  53-55  the  logarithms  of  the  functions  of  angles 
between  1°  and  2°,  or  between  88°  and  89°,  are  given  in  the  manner 
employed  on  pages  50-52.  These  pages  should  be  used  if  the  angle 
lies  between  these  limits,  and  if  not  onlj^  degrees  and  minutes  but 
degrees,  minutes,  and  multiples  of  10"  are  given  or  required. 

When  the  angle  is  between  0°  and  2°,  or  88°  and  90°,  and  a greater  degree 
of  accuracy  is  desired  than  that  given  by  the  table,  interpolation  may  be  em- 
ployed with  some  degree  of  safety  ; but  for  these  angles  interpolation  does  not 
always  give  true  results,  and  it  is  better  to  use  Table  VII. 


INTEODUCTION 


11 


33.  Illustrative  Problems.  The  following  problems  illustrate  the 
use  of  Table  VI  for  small  angles  : 

1.  Find  log  tan  0°  2'  47",  and  log  cos  89°  37'  20". 

log  tan  0°  2'  47"  = log  sin  0°  2'  47"  = 6.90829  — 10.  Page  49 

log  cos  89°  37'  20"  = 7.81911  - 10.  Page  51 

2.  Find  log  cot  0°  2'  15". 

10  - 10 

log  tan  0°  2' 15"=  6.81591-10  Page  49 

Therefore  log  cot  0°  2' 15"=  3.18409 

3.  Find  log  tan  89°  38'  30". 

10  - 10 

log  cot  89°  38'  30"  = 7.79617-  10  Page  51 

Therefore  log  tan  89°  38'  30"  = 2.20383 

4.  Find  x when  log  tan  x = 6.92090  — 10. 

The  nearest  log  tan  is  6.92110  — 10  (page  49),  and  the  angle  corresponding 
to  this  value  of  log  tan  is  0°  2'  52". 

6.  Find  x when  log  cos  x = 7.70240  — 10. 

The  nearest  log  cos  is  7.70261  — 10.  Page  50 

The  corresponding  angle  for  this  value  is  89°  42'  40". 

6.  Find  x when  log  cot  x = 2.37368. 

This  log  cot  is  not  contained  in  the  table. 

The  colog  cot  = 7.62632  — 10  = log  tan. 

The  log  tan  in  the  table  nearest  to  this  is  (page  50)  7.62510  — 10,  and  the 
angle  corresponding  to  this  value  of  log  tan  is  0°  14'  30". 

34.  Angles  between  90°  and  360°.  If  an  angle  x is  between  90°  and 
360°,  it  follows,  from  formulas  established  in  trigonometry,  that, 

Between  90°  and  180°  Between  180°  and  270° 

log  sin  X = log  sin  (180°  — x)  log  sin  x = log  sin  (x  — 180°)„ 

log  cos  X = log  cos  (180°  — a;)„  log  cos  x = log  cos  (x  — 180°)„ 

log  tan  X = log  tan  (180°  — x)„  log  tan  x — log  tan  (x  — 180°) 

log  cot  X = log  cot  (180°  — x)„  log  cot  x = log  cot  (x  — 180°) 

Between  270°  and  360° 
log  sin  X = log  sin  (360°  — *)„ 
log  cos  X = log  cos  (360°  — a:) 
log  tan  X = log  tan  (360°  — a:)„ 
log  cot  X = log  cot  (360°  — a:)„ 

In  these  formulas  the  subscript  n means  that  the  function  is  negative. 
The  logarithm  of  a negative  number  is  imaginary,  so  we  have  to  take  the  loga- 
rithm of  the  number  as  if  it  were  positive ; but  when  we  find  the  function  itself 
we  must  treat  it  as  negative. 


12 


TABLES 


TABLE  VII 


35.  Nature  of  Table  VII.  This  table  (page  78)  must  be  used  when 
great  accuracy  is  desired  in  working  with  angles  between  0°  and  2° 
or  between  88°  and  90°. 

The  values  of  and  T are  such  that  when  the  angle  a is 

expressed  in  seconds,  , . , „ 

S = log  sin  a — log  a , 

T = log  tan  a — log  a". 

Hence  follow  the  formulas  given  on  page  78. 

The  values  of  S and  T are  printed  with  the  characteristic  10  too 
large,  and  in  using  them  — 10  must  always  be  annexed. 


36.  Illustrative  Problems.  The  following  problems  illustrate  the 
use  of  Table  VII  for  angles  near  0°  or  90° : 


1.  Find  log  sin  0°  58' 17". 

0°  58'  17"  = 3497" 
log  3497  = 3.54370 

S = 4,68555  - 10 
log  sin  0°  58'  17"  = 8.22925  - 10 

2.  Find  log  cos  88°  26'  41.2". 
90°  - 88°  26'  41.2"  = 1°  33'  18.8" 

= 5598.8" 
log  5598.8  = 3.74809 

S = 4,68552  - 10 
log  cos  88°  26'  41.2"  = 8.43361  - 10 
This  is  nearer  than  by  page  54. 

5.  Find  x when  log  sin  x — 


3.  Find  log  tan  0°  52'  47.5". 

0°  52' 47.5"  = 3167.-5" 
log  3167.5  = 3.50072 

T = 4.68.561  - 10 
log  tan  0°  52'  47.5"  = 8.18633  - 10 

4.  Find  log  tan  89°  54'  37.362". 

90°  - 89°  54'  37.362"  = 0°  5'  22.638" 

= 322.638" 
log  322.638  = 2.50871 

T = 4.68558  - 10 
log  cot  89°  54'  37.362"  = 7.19429  - 10 
log  tan  89°  54'  37.362"  = 2.80571 

;o6-io. 


Subtracting, 

and 


6.72306-  10 
S = 4.68557  - 10 

2.03749  = log  109.015 

109.015"  = 0°  1'  49.015" 


6.  Find  X when  log  cot  x = 1.67604. 

colog  cot  j;  = 8.32396  — 10 
T = 4.68564  - 10 

Subtracting,  3.63832  = log  4348.3 

and  4348.3"  = 1°  12'  28.3" 


7.  Find  x when  log  tan  x = 1.55407. 

colog  tan  X = 8.44593  — 10 
T = 4.68569  - 10 

Subtracting,  3.76024  = log  5757.6 

5757.6"  = 1°  35'  57.6" 

and  90°  - 1°  35'  57.6"  = 88°  24'  2.4" 

Therefore  the  angle  required  is  88°  24'  2.4". 


INTEODUCTION 


13 


TABLE  VIII 

37.  Nature  of  Table  VIII.  This  table  (pages  79-101)  contains  the 
natural  sines,  cosines,  tangents,  and  cotangents  of  angles  from  0°  to 
90°,  at  intervals  of  1'.  If  greater  accuracy  is  desired,  interpolation 
may  be  employed. 

The  table  is  arranged  on  a plan  similar  to  that  used  in  Table  VI. 

Angles  from  0°  to  44°  are  listed  at  the  top  of  the  pages,  the  minutes  being 
read  downwards  in  the  left-hand  column.  Angles  from  45°  to  89°  are  listed  at 
the  bottom,  the  minutes  being  read  upwards  in  the  right-hand  column. 

The  names  of  the  functions  at  the  top  of  the  columns  are  to  be  used  in  read- 
ing downwards,  and  those  at  the  bottom  are  to  be  used  in  reading  upwards. 

38.  Illustrative  Problems.  The  following  problems  illustrate  the 
use  of  Table  VIII : 

1.  Find  sin  5°  29'. 

We  find  directly  from  the  table  (page  82)  that 

sin  5°  29'  = 0.0966 

2.  Find  cot  78°  18'. 

We  find  directly  from  the  table  (page  85)  that 

cot  78°  18'  = 0.2071 

3.  Find  cos  42°  7'  30". 

From  the  table  (page  100),  cos  42°  7'  = 0.7418 

Tabular  difference  = 0.0002. 

of  this  difference  = 0.0001 

Since  the  cosine  is  decreasing,  we  subtract. 

.-.  cos  42°  7'  30"  = 0.7417 

4.  Find  tan  75°  35'  25". 

From  the  table  (page  86),  tan  76°  35'  = 3.8900 

Tabular  difference  = 0.0047. 

|-5  of  this  difference  = 0.00196  = 0.0020 

Since  the  tangent  is  increasing,  we  add. 

.-.  tan  76°  35' 25"  = 3.8920 

TABLE  IX 

39.  Nature  of  Table  IX.  This  table  converts  degrees  to  radians, 
and  also  degrees  and  parts  of  a degree  indicated  by  10',  20',  30',  40', 
and  50'. 

40.  Illustrative  Problems.  The  following  problems  illustrate  the 
use  of  Table  IX  : 

1.  Express  62°  as  radians. 

From  the  table,  62°  = 1.0821  radians. 

2.  Express  82°  40'  as  radians. 

From  the  table,  82°  40'  = 1.4428  radians. 


14 


TABLES 


TABLE  X 


41.  Nature  of  Table  X.  This  table  converts  minutes  to  thousandths 
of  a degree,  and  seconds  to  ten-thousandths  of  a degree,  this  being 
accurate  enough  for  all  the  purposes  of  elementary  trigonometry. 
It  also  converts  thonsandths  of  a degree,  from  0.001°  to  0.009°,  to 
seconds ; and  hundredths  of  a degree  to  minutes  and  seconds,  so 
that  a computer  who  has  the  decimal  divisions  of  an  angle  given  can 
easily  find  the  sexagesimal  equivalent. 

Table  X thus  provides  for  using  the  decimal  divisions  of  the 
degree  instead  of  the  ancient  sexagesimal  division  into  minutes 
and  seconds. 

There  seems  to  be  little  doubt  that  the  cumbersome  division  of  the  degree 
into  60  minutes,  and  the  minute  into  60  seconds, will  disappear  in  due  time,  by  the 
introduction  either  of  the  grade  (0.01  of  a ijght  angle)  divided  decimally  or  of 
decimal  divisions  of  the  degree.  At  present,  however,  it  must  be  remembered 
that  our  instruments  for  the  measure  of  angles  are  generally  arranged  upon 
the  sexagesimal  scale,  and  that  we  can  serve  the  new  system  best  by  making 
the  change  gradually.  It  is  of  first  importance  that  the  student  shall  learn  how 
to  use  the  common  sexagesimal  system. 

42.  Illustrative  Problems.  The  following  problems  illustrate  the 
use  of  the  table : 


1.  Find  sin  21.34°. 

By  Table  X,  0.34°  = 20'  24" 

Hence  we  have  to  find  sin  21°  20'  24". 

By  Table  VIII,  sin  21°  20'  24"  = 0.36390 


2.  Find  log  tan  15.963°. 

By  Table  X, 
and 

By  Table  V, 

3.  Find  cos  63.72°. 


0.96°  = 57'  36" 

0.003° 11" 
15.963°=  15°  57' 47" 
log  tan  15°  57'  47"  = 9.45644  — 10 


By  Table  X,  0. 72°  = 43'  12" 

Hence  we  have  to  find  cos  63°  43'  12". 

By  Table  VIII,  cos  63°  43'  12"  = 0.4427 


4.  Find  tan  68.651°. 

By  Table  X,  0.651°  = 39'  4" 

Hence  we  have  to  find  tan  68°  39'  4". 

By  Table  VIII,  tan  68°  39'  4"  = 2.5538 

6.  Find  log  cot  56.388°. 

By  Table  X,  0.388°  = 23'  17" 

Hence  we  have  to  find  log  cot  56°  23'  17". 

By  Table  VIII,  log  cot  56°  23'  1 7"  = 9.82262 


INTEODUCTION 


15 


EXERCISE 


Using  Table  I,  find  the  logarithms  of  the  following : 


1.  75. 

7.  57.8. 

13.  0.725. 

19.  8. 

25.  140. 

2.  96. 

8.  42.6. 

14.  7.250. 

20.  0.8. 

26.  141. 

3.  37. 

9.  93.9. 

15.  72.50. 

21.  0.08. 

27.  14.2. 

4.  423. 

10.  4.27. 

16.  24.3. 

22.  0.008. 

28.  1.43. 

5.  568. 

11.  6.42. 

17.  2.43. 

23.  8.08. 

29.  0.144. 

6.  647. 

12.  7.53. 

18.  0.243. 

24.  8.80. 

30.  0.145. 

Using  Table  f find  the  antilogarithms  of  the  following  : 

31.  1.4771. 

37.  2.5988. 

43.  1.9510. 

49. 

1.9518. 

32.  0.9031. 

38.  1.6590. 

44.  0.9607. 

50. 

2.8978. 

33.  1.7076. 

39.  4.6749. 

45.  3.9753. 

51. 

0.9335. 

34.  1.9031. 

40.  3.9595. 

46.  2.6196. 

52. 

4.8460. 

35.  1.9345. 

41.  0.9581. 

47.  0.6360. 

53. 

1.3714. 

36.  0.8451. 

42.  2.8494. 

48.  2.6640. 

54. 

2.4448. 

Using  Table  I,  find  the  logarithms  of  the  following : 


55.  log  sin  29°. 

56.  log  cos  42°. 

57.  log  tan  51°. 

58.  log  cot  20°. 

59.  log  sin  45°. 

60.  log  cos  45°. 


61.  log  sin  6°  10'. 

62.  log  cos  7°  20'. 

63.  log  tan  5°  30'. 

64.  log  cot  8°  50'. 

65.  log  sin  45°  10'. 

66.  log  cos  44°  80'. 


67.  log  sin  20°  10'. 

68.  log  cos  42°  20'. 

69.  log  tan  37°  50'. 

70.  log  cot  82°  40'. 

71.  log  sin  22°  30'. 

72.  log  tan  81°  10'. 


Using  Table  f find  the  value  of  x in  the  following : 

73.  log  sin  X = 9.7861. 

74.  log  sin  X = 9.9116. 

75.  log  tan  a;  9.9772. 


76.  log  tan  a;  = 9.8771. 

77.  log  cos  X = 9.9089. 

78.  log  cot  X = 10.0711, 


79.  log  sin  X — 9.8058. 

80.  log  cos  X = 9.9252. 

81.  log  cos  X = 9.9101. 

82.  log  tan  X = 8.9118. 

83.  log  tan  X — 9.0093. 

84.  log  cot  X = 10.1944. 


Using  Table  Ilf  find  the  logarithms  of  the  following  : 

85.  1475.  88.  564.8.  91.  29.37.  94.  0.4236. 

86.  2836.  89.  392.7.  92.  42.86.  95.  0.09873. 

87.  4293.  90.  586.4.  93.  53.91.  96.  487.48. 


Using  Table  Ilf  find  the  antilogarithms  of  the  following  : 

97.  2.02078.  100.  0.82756.  103.  2.95873.  106.  0.70804. 

98.  3.55967.  101.  1.82988.  104.  3.81792.  107.  2.34404. 

99.  1.75686.  102.  2.96052.  105.  1.82725.  108.  3.35054. 


16 


TABLES 


Using  Table  VI,  find  the  following  logarithms  : 


109.  log  sin  10°. 

110.  log  sin  30°. 

111.  log  sin  60°. 

112.  log  sin  79°. 

113.  log  cos  87°. 

114.  log  tan  33°. 

115.  log  cot  72°. 


116.  log  sin  1' 51". 

117.  log  tan  37' 50". 

118.  log  cos  1°  19'. 

119.  log  cot  88°  24'. 

120.  log  sin  19°  37'. 

121.  log  cos  72°  43'. 

122.  log  cot  88°  18'. 


123.  log  sin  10'  37". 

124.  log  cot  67°  42'. 

125.  log  cos  32°  36' 10". 

126.  log  tan  73°  42' 15". 

127.  log  sin  15°  15' 15". 

128.  log  COS  29°  32' 40". 

129.  log  cot  78°  33' 25". 


Using  Table  VI,  find  the  value  of  x in  the  following : 


130.  log  since  = 9.52563. 

131.  log  cotcc  = 9.57658. 

132.  log  cos  X = 9.73435. 


133.  log  sin  X = 9.93386. 

134.  log  cot  cc  = 9.75837. 

135.  log  cos  X = 9.99843. 


Using  Table  IV,  find  the  following : 

136.  0.8  of  37.  137.  0.6  of  79.  138.  0.7  of  68.  139.  0.9  of  29. 


Using  Table  V,  find  the  folloiving : 

140.  log  4 7T.  141.  log  142.  log  57.2958°.  143.  log  ^^5. 

Using  Table  VH,  find  the  following  : 

144.  log  sin  57".  145.  log  sin  48".  146.  log  tan  89°  58' 10". 

Using  Table  V,  find  the  following  : 


147.  2 77-87. 

148.  IT  • 7 

149. 

Q ^0 

150.  — 

2 77 

4 77 

Using  Table  FZZZ^  find  the  following : 

151.  sin  10°  17'. 

155.  COS  46°  38'. 

159. 

cot  1°  52'. 

152.  sin  37°  40'. 

156.  cos  78°  19'. 

160. 

cot  63°  48'. 

153.  sin  68°  10'. 

157.  tan  16°  29'. 

161. 

cot  10°  9'  10". 

154.  cos  10°  39'. 

158.  tan  88°  8'. 

162. 

cot  5°  17'  8". 

163.  The  angles  whose  sines  are  0.5113  and  0.7801. 

Using  Table  IX,  express  the  following  : 

164.  52°  40'  as  radians.  165.  0.8116  radians  as  degrees. 

Using  Table  X,  express  the  following : 

166.  31' as  a decimal  of  a degree.  167.  0.96°  as  minutes  and  seconds. 


17 


TABLE  I 


FOUR-PLACE  MANTISSAS 
OF  THE  COMMON  LOGARITHMS  OF 
INTEGERS  FROM  1 TO  1000 
AND  OF  THE  TRIGONOMETRIC  FUNCTIONS 


On  this  page  the  logarithms  of  integers  from  1 to  100  are  given  in 
full,  with  characteristics  as  well  as  mantissas.  On  account  of  the  great 
differences  between  the  successive  mantissas,  interpolation  cannot  safely  be 
employed  on  this  page.  On  pages  18  and  19  are  given  the  mantissas  of 
numbers  from  100  to  1000,  and  on  pages  20-23  the  logarithms  of  trigono- 
metric functions. 


1-100 


N 

log 

N 

log 

N 

log 

N 

log 

N 

log 

1 

0.  0000 

21 

1.  3222 

41 

1.6128 

61 

1.  7853 

81 

1.  9085 

2 

0.  3010 

22 

1.  3424 

42 

1.  6232 

62 

1.  7924 

82 

1.  9138 

3 

0.  4771 

23 

1.3617 

43 

1.  6335 

63 

1.  7993 

83 

1.9191 

4 

0.  6021 

24 

1.  3802 

44 

1.6435 

64 

1.  8062 

84 

1.  9243 

S 

0.  6990 

25 

1.  3979 

45 

1.  6532 

65 

1.  8129 

85 

1.  9294 

6 

0.  7782 

26 

1.  4150 

46 

1.  6628 

66 

1.8195 

86 

1.  9345 

7 

0.  8451 

27 

1.  4314 

47 

1.  6721 

67 

1.  8261 

87 

1.  9395 

8 

0.  9031 

28 

1.4472 

48 

1.6812 

68 

1.  8325 

88 

1.  9445 

9 

0.  9542 

29 

1.  4624 

49 

1.  6902 

69 

1.  8388 

89 

1.9494 

10 

1.  0000 

30 

1.4771 

50 

1.  6990 

70 

1.8451 

90 

1.  9542 

11 

1.  0414 

31 

1.  4914 

51 

1.  7076 

71 

1.8513 

91 

1.  9590 

12 

1.  0792 

32 

1.  5051 

52 

1.  7160 

72 

1.8573 

92 

1.  9638 

13 

1. 1139 

33 

1.  5185 

53 

1.  7243 

73 

1.  8633 

93 

1.  9685 

14 

1.  1461 

34 

1.5315 

54 

1.  7324 

74 

1.8692 

94 

1.  9731 

IS 

1. 1761 

35 

1.5441 

55 

1.  7404 

75 

1.8751 

95 

1.9777 

16 

1.  2041 

36 

1.  5563 

56 

1.  7482 

76 

1.  8808 

96 

1.  9823 

17 

1.  2304 

37 

1.  5682 

57 

1.  7559 

77 

1.  8865 

97 

1.  9868 

18 

1.  2553 

38 

1.  5798 

58 

1.  7634 

78 

1.  8921 

98 

1.  9912 

19 

1.  2788 

39 

1.5911 

59 

1.  7709 

79 

1.  8976 

99 

1.  9956 

20 

1.  3010 

40 

1.  6021 

60 

1.  7782 

80 

1.  9031 

100 

2.  0000 

N 

log 

N 

log 

N 

log 

N 

log 

N 

log 

1-100 


18 


100-500 


Each  mantissa  should  be  preceded  by  a decimal  point,  and  the  proper 
characteristic  should  be  written. 

On  account  of  the  great  differences  between  the  successive  mantissas 
in  the  first  ten  rows,  interpolation  should  not  be  employed  in  that  part  of 
the  table.  Table  III  should  be  used  in  this  case.  In  general,  an  error  of 
one  unit  may  appear  in  the  last  figure  of  any  interpolated  value. 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

lO 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

62‘13 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100-500 


500-1000 


19 


N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

60 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

56 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

■ 9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

86 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

100 

0000 

0004 

0009 

0013 

0017 

0022 

0026 

0030 

0035 

0039 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

500-1000 


20 


LOGARITHMS  OF  SINES 


o 

O' 

lO' 

20' 

30' 

40' 

50' 

60' 

O 

o 

— 00 

7.4637 

7.7648 

7.9408 

8.0658 

8.1627 

8.2419 

89 

1 

8.2419 

8.3088 

8.3668 

8.4179 

4637 

5050 

5428 

88 

2 

5428 

5776 

6097 

6397 

6677 

6940 

7188 

87 

3 

7188 

7423 

7645 

7857 

8059 

8251 

8436 

86 

4 

8436 

8613 

8783 

8946 

9104 

8.9256 

8.9403 

8o 

5 

8.9403 

8.9545 

8.9682 

8.9816 

8.9945 

9.0070 

9.0192 

84 

6 

9.0192 

9.0311 

9.0426 

9.0539 

9.0648 

0755 

0859 

83 

7 

0859 

0961 

1060 

1157 

1252 

1345 

1436 

82 

8 

1436 

1525 

1612 

1697 

1781 

1863 

1943 

81 

9 

1943 

2022 

2100 

2176 

2251 

2324 

2397 

80 

10 

9.2397 

9.2468 

9.2538 

9.2606 

9.2674 

9.2740 

9.2806 

79 

11 

2806 

2870 

2934 

2997 

3058 

3119 

3179 

78 

12 

3179 

3238 

3296 

3353 

3410 

3466 

3521 

77 

13 

3521 

3575 

3629 

3682 

3734 

3786 

3837 

76 

14 

3837 

3887 

3937 

3986 

4035 

4083 

4130 

75 

15 

9.4130 

9.4177 

9.4223 

9.4269 

9.4314 

9.4359 

9.4403 

74 

16 

4403 

4447 

4491 

4533 

4576 

4618 

4659 

73 

17 

4659 

4700 

4741 

4781 

4821 

4861 

4900 

72 

18 

4900 

4939 

4977 

5015 

5052 

5090 

5126 

71 

19 

5126 

5163 

5199 

5235 

5270 

5306 

5341 

70 

20 

9.5341 

9.5375 

9.5409 

9.5443 

9.5477 

9.5510 

9.5543 

69 

21 

5543 

5576 

5609 

5641 

5673 

5704 

5736 

68 

22 

5736 

5767 

5798 

5828 

5859 

5889 

5919 

67 

23 

5919 

5948 

5978 

6007 

6036 

6065 

6093 

66 

24 

6093 

6121 

6149 

6177 

6205 

6232 

6259 

Co 

25 

9.6259 

9.6286 

9.6313 

9.6340 

9.6366 

9.6392 

9.6418 

64 

26 

6418 

6444 

6470 

6495 

6521 

6546 

6570 

63 

27 

6570 

6595 

6620 

6644 

6668 

6692 

6716 

62 

28 

6716 

6740 

6763 

6787 

6810 

6833 

6856 

61 

29 

6856 

6878 

6901 

6923 

6946 

6968 

6990 

60 

30 

9.6990 

9.7012 

9.7033 

9.7055 

9.7076 

9.7097 

9.7118 

59 

31 

7118 

7139 

7160 

7181 

7201 

7222 

7242 

58 

32 

7242 

7262 

7282 

7302 

7322 

7342 

7361 

57 

33 

7361 

7380 

7400 

7419 

7438 

7457 

7476 

56 

34 

7476 

7494 

7513 

7531 

7550 

7568 

7586 

55 

35 

9.7586 

9.7604 

9.7622 

9.7640 

9.7657 

9.7675 

9.7692 

54 

36 

7692 

7710 

7727 

7744 

7761 

7778 

7795 

53 

37 

7795 

7811 

7828 

7844 

7861 

7877 

7893 

52 

38 

7893 

7910 

7926 

7941 

7957 

7973 

7989 

51 

39 

7989 

8004 

8020 

8035 

8050 

8066 

8081 

50 

40 

9.8081 

9.8096 

9.8111 

9.8125 

9.8140 

9.8155 

9.8169 

49 

41 

8169 

8184 

8198 

8213 

8227 

8241 

8255 

48 

42 

8255 

8269 

8283 

8297 

8311 

8324 

8338 

47 

43 

8338 

8351 

8365 

8378 

8391 

8405 

8418 

46 

44 

9.8418 

9.8431 

9.8444 

9.8457 

9.8469 

9.S4S2 

9.8495 

A5 

0 

60' 

60' 

40' 

30' 

20' 

10' 

O' 

O 

LOGARITHMS  OF  COSINES 


LOGARITHMS  OF  COSINES 


21 


o 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

o 

o 

10.0000 

10.0000 

10.0000 

10.0000 

10.0000 

10.0000 

9.9999 

89 

1 

9.9999 

9.9999 

9.9999 

9.9999 

9.9998 

9.9998 

9997 

88 

2 

9997 

9997 

9996 

9996 

9995 

9995 

9994 

87 

3 

9994 

9993 

9993 

9992 

9991 

9990 

9989 

86 

4 

9989 

9989 

9988 

9987 

9986 

9985 

9983 

86 

6 

9.9983 

9.9982 

9.9981 

9.9980 

9.9979 

9.9977 

9.9976 

84 

6 

9976 

9975 

9973 

9972 

9971 

9969 

9968 

83 

7 

9968 

9966 

9964 

9963 

9961 

9959 

9958 

82 

8 

9958 

9956 

9954 

9952 

9950 

9948 

9946 

81 

9 

9946 

9944 

9942 

9940 

9938 

9936 

9934 

80 

10 

9.9934 

9.9931 

9.9929 

9.9927 

9.9924 

9.9922 

9.9919 

79 

11 

9919 

9917 

9914 

9912 

9909 

9907 

9904 

78 

12 

9904 

9901 

9899 

9896 

9893 

9890 

9887 

77 

13 

9887 

9884 

9881 

9878 

9875 

9872 

9869 

76 

14 

9869 

9866 

9863 

9859 

9856 

9853 

9849 

76 

15 

9.9849 

9.9846 

9.9843 

9.9839 

9.9836 

9.9832 

9.9828 

74 

16 

9828 

9825 

9821 

9817 

9814 

9810 

9806 

73 

17 

9806 

9802 

9798 

9794 

9790 

9786 

9782 

72 

18 

9782 

9778 

9774 

9770 

9765 

9761 

9757 

71 

19 

9757 

9752 

9748 

9743 

9739 

9734 

9730 

70 

20 

9.9730 

9.9725 

9.9721 

9.9716 

9.9711 

9.9706 

9.9702 

69 

21 

9702 

9697 

9692 

9687 

9682’ 

9677 

9672 

68 

22 

9672 

9667 

9661 

9656 

9651 

9646 

9640 

67 

23 

9640 

9635 

9629 

9624 

9618 

9613 

9607 

66 

24 

9607 

9602 

9596 

9590 

9584 

9579 

9573 

65 

26 

9.9573 

9.9567 

9.9561 

9.9555 

9.9549 

9.9543 

9.9537 

64 

26 

9537 

9530 

9524 

9518 

9512 

9505 

9499 

63 

27 

9499 

9492 

9486 

9479 

9473 

9466 

9459 

62 

28 

9459 

9453 

9446 

9439 

9432 

9425 

9418 

61 

29 

9418 

9411 

9404 

9397 

9390 

9383 

9375 

60 

30 

9.9375 

9.9368 

9.9361 

9.9353 

9.9346 

9.9338 

9.9331 

59 

31 

9331 

9323 

9315 

9308 

9300 

9292 

9284 

58 

32 

9284 

9276 

9268 

9260 

9252 

9244 

9236 

57 

33 

9236 

9228 

9219 

9211 

9203 

9194 

9186 

56 

34 

9186 

9177 

9169 

9160 

9151 

9142 

9134 

65 

35 

9.9134 

9.9125 

9.9116 

9.9107 

9.9098 

9.9089 

9.9080 

54 

36 

9080 

9070 

9061 

9052 

9042 

9033 

9023 

53 

37 

9023 

9014 

9004 

8995 

8985 

8975 

8965 

52 

38 

8965 

8955 

8945 

8935 

8925 

8915 

8905 

51 

39 

8905 

8895 

8884 

8874 

8864 

8853 

8843 

50 

40 

9.8843 

9.8832 

9.8821 

9.8810 

9.8800 

9.8789 

9.8778 

49 

41 

8778 

8767 

8756 

8745 

8733 

8722 

8711 

48 

42 

8711 

8699 

8688 

8676 

8665 

8653 

8641 

47 

43 

8641 

8629 

8618 

8606 

8594 

8582 

8569 

46 

44 

9.8569 

9.8557 

9.8545 

9.8532 

9.8520 

9.8507 

9.8495 

45 

0 

60' 

50' 

40' 

30' 

20' 

lO' 

O' 

O 

LOGARITHMS  OF  SINES 


22 


LOGAKITHMS  OF  TANGENTS 


o 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

0 

o 

— 00 

7.4637 

7.7648 

7.9409 

8.0658 

8.1627 

8.2419 

89 

1 

8.2419 

8.3089 

8.3669 

8.4181 

4638 

5053 

5431 

88 

2 

5431 

5779 

6101 

6401 

6682 

6945 

7194 

87 

3 

7194 

7429 

7652 

7865 

8067 

8261 

8446 

86 

4 

8446 

8624 

8795 

8960 

9118 

8.9272 

8.9420 

85 

5 

8.9420 

8.9563 

8.9701 

8.9836 

8.9966 

9.0093 

9.0216 

84 

6 

9.0216 

9.0336 

9.0453 

9.0567 

9.0678 

0786 

0891 

83 

7 

0891 

0995 

1096 

1194 

1291 

1385 

1478 

82 

8 

1478 

1569 

1658 

1745 

1831 

1915 

1997 

81 

9 

1997 

2078 

2158 

2236 

2313 

2389 

2463 

80 

lO 

9.2463 

9.2536 

9.2609 

9.2680 

9.2750 

9.2819 

9.2887 

79 

11 

2887 

2953 

3020 

3085 

3149 

3212 

3275 

78 

12 

3275 

3336 

3397 

3458 

3517 

3576 

3634 

77 

13 

3634 

3691 

3748 

3804 

3859 

3914 

3968 

76 

14 

3968 

4021 

4074 

4127 

4178 

4230 

4281 

75 

16 

9.4281 

9.4331 

9.4381 

9.4430 

9.4479 

9.4527 

9.4575 

74 

16 

4575 

4622 

4669 

4716 

4762 

4808 

4853 

73 

17 

4853 

4898 

4943 

4987 

5031 

5075 

5118 

72 

18 

5118 

5161 

5203 

5245 

5287 

5329 

5370 

71 

19 

5370 

5411 

5451 

5491 

5531 

5571 

5611 

70 

20 

9.5611 

9.5650 

9.5689 

9.5727 

9.5766 

9.5804 

9.5842 

69 

21 

5842 

5879 

■ 5917 

5954 

5991 

6028 

6064 

68 

22 

6064 

6100 

6136 

6172 

6208 

6243 

6279 

67 

23 

6279 

6314 

6348 

6383 

6417 

6452 

6486 

66 

24 

6486 

6520 

6553 

6587 

6620 

6654 

6687 

65 

26 

9.6687 

9.6720 

9.6752 

9.6785 

9.6817 

9.6850 

9.6882 

64 

26 

6882 

6914 

6946 

6977 

7009 

7040 

7072 

63 

27 

7072 

7103 

7134 

7165 

7196 

7226 

7257 

62 

28 

7257 

7287 

7317 

7348 

7378 

7408 

7438 

61 

29 

7438 

7467 

7497 

7526 

7556 

7585 

7614 

60 

30 

9.7614 

9.7644 

9.7673 

9.7701 

9.7730 

9.7759 

9.7788 

59 

31 

7788 

7816 

7845 

7873 

7902 

7930 

7958 

58 

32 

7958 

7986 

8014 

8042 

8070 

8097 

8125 

57 

33 

8125 

8153 

8180 

8208 

8235 

8263 

8290 

56 

34 

8290 

8317 

8344 

8371 

8398 

8425 

8452 

55 

36 

9.8452 

9.8479 

9.8506 

9.8533 

9.8559 

9.8586 

9.8613 

54 

36 

8613 

8639 

8666 

8692 

8718 

8745 

8771 

53 

37 

8771 

8797 

8824 

8850 

8876 

8902 

8928 

52 

38 

8928 

8954 

8980 

9006 

9032 

9058 

9084 

51 

39 

9084 

9110 

9135 

9161 

9187 

9212 

9238 

50 

40 

9.9238 

9.9264 

9.9289 

9.9315 

9.9341 

9.9366 

9.9392 

49 

41 

9392 

9417 

9443 

9468 

9494 

9519 

9544 

48 

42 

9544 

9570 

9595 

9621 

9646 

9671 

9697 

47 

43 

9697 

9722 

9747 

9772 

9798 

9823 

9.9S4S 

46 

44 

9.9848 

9.9874 

9.9899 

9.9924 

9.9949 

9.9975 

10.0000 

4:5 

o 

60' 

50' 

40' 

30' 

20' 

lO' 

O' 

O 

LOGAKITHMS  OF  COTANGENTS 


LOGARITHMS  OF  COTANGENTS 


23 


o 

O' 

lO' 

20' 

30' 

40' 

60' 

60' 

o 

0 

00 

12.5363 

12.2352 

12.0591 

11.9342 

11.8373 

11.7581 

89 

1 

11.7581 

11.6911 

11.6331 

11.5819 

5362 

4947 

4569 

88 

2 

4569 

4221 

3899 

3599 

3318 

3055 

2806 

87 

3 

2806 

2571 

2348 

2135 

1933 

1739 

1554 

86 

4 

1554 

1376 

1205 

1040 

0882 

11.0728 

11.0580 

85 

5 

11.0580 

11.0437 

11.0299 

11.0164 

11.0034 

10.9907 

10.9784 

84 

6 

10.9784 

10.9664 

10.9547 

10.9433 

10.9322 

9214 

9109 

83 

7 

9109 

9005 

8904 

8806 

8709 

8615 

8522 

82 

8 

8522 

8431 

8342 

8255 

8169 

8085 

8003 

81 

9 

8003 

7922 

7842 

7764 

7687 

7611 

7537 

80 

lO 

10.7537 

10.7464 

10.7391 

10.7320 

10.7250 

10.7181 

10.7113 

79 

11 

7113 

7047 

6980 

6915 

6851 

6788 

6725 

78 

12 

6725 

6664 

6603 

6542 

6483 

5424 

6366 

77 

13 

6366 

6309 

6252 

6196 

6141 

6086 

6032 

76 

14 

6032 

5979 

5926 

5873 

5822 

5770 

5719 

75 

15 

10.5719 

10.5669 

10.5619 

10.5570 

10.5521 

10.5473 

10.5425 

74 

16 

5425 

5378 

5331 

5284 

5238 

5192 

5147 

73 

17 

5147 

5102 

5057 

5013 

4969 

4925 

4882 

72 

18 

4882 

4839 

4797 

4755 

4713 

4671 

4630 

71 

19 

4630 

4589 

4549 

4509 

4469 

4429 

4389 

70 

20 

10.4389 

10.4350 

10.4311 

10.4273 

10.4234 

10.4196 

10.4158 

69 

21 

4158 

4121 

4083 

4046 

4009 

3972 

3936 

68 

22 

3936 

3900 

3864 

3828 

3792 

3757 

3721 

67 

23 

3721 

3686 

3652 

3617 

3583 

3548 

3514 

66 

24 

3514 

3480 

3447 

3413 

3380 

3346 

3313 

65 

26 

10.3313 

10.3280 

10.3248 

10.3215 

10.3183 

10.3150 

10.3118 

64 

26 

3118 

3086 

3054 

3023 

2991 

2960 

2928 

63 

27 

2928 

2897 

2866 

2835 

2804 

2774 

2743 

62 

28 

2743 

2713 

2683 

2652 

2622 

2592 

2562 

61 

29 

2562 

2533 

2503 

2474 

2444 

2415 

2386 

60 

30 

10.2386 

10.2356 

10.2327 

10.2299 

10.2270 

10.2241 

10.2212 

59 

31 

2212 

2184 

2155 

2127 

2098 

2070 

2042 

58 

32 

2042 

2014 

1986 

1958 

1930 

1903 

1875 

57 

33 

1875 

1847 

1820 

1792 

1765 

1737 

1710 

56 

34 

1710 

1683 

1656 

1629 

1602 

1575 

1548 

55 

35 

10.1548 

10.1521 

10.1494 

10.1467 

10.1441 

10.1414 

10.1387 

54 

36 

1387 

1361 

1334 

1308 

1282 

1255 

1229 

53 

37 

1229 

1203 

1176 

1150 

1124 

1098 

1072 

52 

38 

1072 

1046 

1020 

0994 

0968 

0942 

0916 

51 

39 

0916 

0890 

0865 

0839 

0813 

0788 

0762 

50 

40 

10.0762 

10.0736 

10.0711 

10.0685 

10.0659 

10.0634 

10.0608 

49 

41 

0608 

0583 

0557 

0532 

0506 

0481 

0456 

48 

42 

0456 

0430 

0405 

0379 

0354 

0329 

0303 

47 

43 

0303 

0278 

0253 

0228 

0202 

0177 

0152 

46 

44 

10.0152 

10.0126 

10.0101 

10.0076 

10.0051 

10.0025 

10.0000 

45 

O 

60' 

60' 

40' 

30' 

20' 

10' 

O' 

O 

LOGARITHMS  OF  TANGENTS 


24 


CIRCLES,  POWERS,  AND  ROOTS 


TABLE 

II 

a 

ird 

d^ 

d^ 

V~d 

Vd 

o 

0.0000 

0.0000 

0 

0 

0.0000 

0.0000 

1 

3.1416 

0.7854 

1 

1 

1.0000 

1.0000 

2 

6.2832 

3.1416 

4 

8 

4142 

2599 

3 

9.4248 

7.0686 

9 

27 

1.7321 

4422 

4 

12.5664 

12.5664 

16 

64 

2.0000 

5874 

5 

15.7080 

19.6350 

25 

125 

2.2361 

1.7100 

6 

18.8496 

28.2743 

36 

216 

,44*  < 

8171 

7 

21.9911 

38.4845 

49 

343 

6458 

1.9129 

8 

25.1327 

50.2655 

64 

512 

2.8284 

2.0000 

9 

28.2743 

63.6173 

81 

729 

3.0000 

0801 

lO 

31.4159 

78.5398 

100 

1,000 

3.1623 

2.1544 

11 

34.5575 

95.0332 

^ 1-21- 

1,331 

^.3166 

' 2240 

12 

37.6991 

113.0973 

144 

1,728 

',,4641 

2894 

13 

40.8407 

132.7323 

169 

2,197 

,6056 

3513 

14 

43.9823 

153.9380 

196 

2,744 

. 7417 

4101 

15 

47.1239 

176.7146 

225 

3,375 

3.8730 

2.4662 

16 

50.2655 

201.0619 

256 

4,096 

4.0000 

5198 

17 

53.4071 

226.9801 

289 

4,913 

1231 

5713 

18 

56.5487 

254.4690 

324 

5,832 

2426 

6207 

19 

59.6903 

283.5287 

361 

6,859  , 

3589 

6684 

20 

62.8319 

314.1593 

400 

8,000 

4.4721 

2.7144 

21 

65.9734 

346.3606 

441 

9,261 

.5826 

7589 

22 

69.1150 

380.1327 

484 

10,648 

6904 

8020 

23 

72.2566 

415.4756 

529 

12,167 

7958 

8439 

24 

75.3982 

452.3893 

576 

13,824 

4.8990 

8845 

25 

78.5398 

490.8739 

625 

15,625 

5.0000 

2.9240 

26 

81.6814 

530.9292 

676 

17,576 

0990 

2.9625 

27 

84.8230 

572.5553 

729 

19,683 

1962 

3.0000 

28 

87.9646 

615.7522 

784 

21,952 

2915 

0366 

29 

91.1062 

660.5199 

841 

24,389 

3852 

0723 

30 

94.2478 

706.8583 

900 

27,000 

5.4772 

3.1072 

31 

97.3894 

754.7676 

961 

29,791 

5678 

1414 

32 

100.5310 

804.2477 

1024 

32,768 

6569 

1748 

33 

103.6726 

855.2986 

1089 

35,937 

7446 

2075 

34 

106.8142 

907.9203 

1156 

39,304 

8310 

2396 

35 

109.9557 

962.1128 

■ 1225 

42,875 

5.9161 

3.2711 

36 

113.0973 

1017.8760 

1296 

46,656 

6.0000 

3019 

37 

116.2389 

1075.2101 

1369 

50,653 

0828 

3322 

38 

119.3805 

1134.1149 

1444 

54,872 

1644 

3620 

39 

122.5221 

1194.5906 

1521 

59,319 

2450 

3912 

40 

125.6637 

1256.6371 

1600 

64,000 

6.3246 

3.4200 

41 

128.8053 

1320.2543 

1681 

68,921 

4031 

4482 

42 

131.9469 

1385.4424 

1764 

74,088 

4807 

4760 

43 

135.0885 

1452.2012 

1849 

79,507 

5574 

5034 

44 

138.2301 

1520.5308 

1936 

85,184 

^6332 

5303 

45 

141.3717 

1590.4313 

2025 

91.125 

6.7082 

3.5569 

46 

144.5133 

1661.9025 

2116 

97,336 

7823 

5830 

47 

147.6549 

1734.9445 

2209 

103,823 

8557 

6088 

48 

150.7964 

1809.5574 

2304 

110,592 

6.9282 

6342 

49 

153.9380 

1885.7410 

2401 

117,649 

7.0000 

6593 

50 

157.0796 

1963.4954 

2500^ 

125,000 

7.0711 

3.6840 

CIECLES,  POWERS,  AND  ROOTS 


25 


CIRCUMFERENCES  AND  AREAS  OF  CIRCLES 
SQUARES,  CUBES,  SQUARE  ROOTS,  CUBE  ROOTS 


d 

ird 

4 

d^ 

Vd 

Vd 

50 

157.0796 

1963.4954 

2500 

125,000 

7.0711 

3.6840 

51 

160.2212 

2042.8206 

2601 

132,651 

1414 

7084 

52 

163.3628 

2123.7166 

2704 

140,608 

2111  . 
2801 

7325 

53 

166.5044 

2206.1834 

2809 

148,877 

7563 

54 

169.6460 

2290.2210 

2916 

157,464  • 

3485 

7798 

55 

172.7876 

2375.8294 

3025 

166,375 

7.4162 

3.8030 

56 

175.9292 

2463.0086 

3136 

175,616 

4833 

8259 

57 

179.0708 

2551.7586 

3249 

185,193 

5498 

8485 

58 

182.2124 

2642.0794 

3364 

195,112 

6158 

8709 

59 

185.3540 

2733.9710 

3481 

205,379 

6811 

8930 

60 

188.4956 

2827.4334 

3600 

216,000 

7.7460 

3.9149 

61 

191.6372 

2922.4666 

3721 

226,981 

8102 

9365 

62 

194.7787 

3019.0705 

3844 

238,328 

8740 

9579 

63 

197.9203 

3117.2453 

3969 

250,047 

7.9373 

3.9791 

64 

201.0619 

3216.9909 

4096 

262,144 

8.0000 

4.0000 

65 

204.2035 

3318.3072 

4225 

274,625 

8.0623 

4.0207 

66 

207.3451 

3421.1944 

4356 

287,496 

1240 

0412 

67 

210.4867 

3525.6524 

4489 

300,763 

1854 

0615 

68 

213.6283 

3631.6811 

4624 

314,432 

2462 

0817 

69 

216.7699 

3739.2807 

4761 

328,509 

3066 

1016 

70 

219.9115 

3848.4510 

4900 

343,000 

8.3666 

4.1213 

71 

223.0531 

3959.1921 

5041 

357,911 

4261 

1408 

72 

226.1947 

4071.5041 

5184 

373,248 

4853 

1602 

73 

229.3363 

4185.3868 

5329 

389,017 

5440 

1793 

74 

232.4779 

4300.8403 

5476 

405,224 

6023 

1983 

75 

235.6194 

4417.8647 

5625 

421,875 

8.6603 

4.2172 

76 

238.7610 

4536.4598 

5776 

438,976 

7178 

2358 

77 

241.9026 

4656.6257 

5929 

456,533 

7750 

2543 

78 

245.0442 

4778.3624 

6084 

474,552 

8318 

2727 

79 

248.1858 

4901.6699 

6241 

493,039 

8882 

2908 

80 

251.3274 

5026.5482 

6400 

512,000 

8.9443 

4.3089 

81 

254.4690 

' 5152.9974 

6561 

531,441 

9.0000 

3267 

82 

257.6106 

5281.0173 

6724 

551,368 

0554 

3445 

83 

260.7522 

5410.6079 

6889 

571,787 

1104 

3621 

84 

263.8938 

5541.7694 

7056 

592,704 

1652 

3795 

85 

267.0354 

5674.5017 

7225 

614,125 

9.2195 

4.3968 

86 

270.1770 

5808.8048 

7396 

636,056 

2736 

4140 

87 

273.3186 

5944.6787 

7569 

658,503 

3274 

4310 

88 

276.4602 

6082.1234 

7744 

681,472 

3808 

4480 

89 

279.6017 

6221.1389 

7921 

704,969 

4340 

4647 

90 

282.7433 

6361.7251 

8100 

729,000 

9.4868 

4.4814 

91 

285.8849 

6503.8822 

8281 

753,571 

5394 

4979 

92 

289.0265 

6647.6101 

8464 

778,688 

5917 

5144 

93 

292.1681 

6792.9087 

8649 

804,357 

6437 

5307 

94 

295.3097 

6939.7782 

8836 

830,584 

6954 

5468 

95 

298.4513 

7088.2184 

9025 

857,375 

9.7468 

4.5629 

96 

301.5929 

7238.2295 

9216 

884,736 

7980 

5789 

97 

304.7345 

7389.8113 

9409 

912,673 

8489 

5947 

98 

307.8761 

7542.9640 

9604 

941,192 

8995 

6104 

99 

311.0177 

7697.6874 

9801 

970,299 

9.9499 

6261 

100 

314.1593 

7853.9816 

10000 

1,000,000 

10.0000 

4.6416 

26 


CIECUMFEEENCES  AND  AEEAS  OF  CIECLES 


If  n = the  radius  of  the  circle,  the  circumference  = 2 tth. 

li  n = the  radius  of  the  circle,  the  area  = 7m^. 

If  n = the  circumference  of  the  circle,  the  radius  = — n. 

2 7T 

If  n = the  circumference  of  the  circle,  the  area  = — v?. 

4 7T 


n 

2TT71 

7T?i2 

1 

“W- 

n 

2 nil 

1 

~ n 

An 

47T 

2n 

4?r 

0 

0.  00 

0.0 

0.  000 

0.00 

50 

314. 16 

7 854 

7.96 

198.  94 

1 

6.  28 

3. 1 

0.  159 

0.  08 

51 

320.  44 

8 171 

8. 12 

206.  98 

2 

12.57 

12.6 

0.318 

0.  32 

52 

326.  73 

8 495 

8.28 

215. 18 

3 

18.  85 

28.3 

0.  477 

0.  72 

53 

333.  01 

8 825 

8.44 

223.  53 

4 

25.13 

50.3 

0.  637 

1.27 

54 

339.  29 

9161 

8.  59 

232.  Oi 

6 

31.42 

78.5 

0.  796 

1.99 

55 

345.  58 

9 503 

8.  75 

240.  72 

6 

37.  70 

113.1 

0.  955 

2.  86 

56 

351.  86 

9 852 

8.91 

249.  55 

7 

43.98 

153.9 

1.  114 

3.90 

57 

358. 14 

10  207 

9. 07 

258.  55 

8 

50.  27 

201. 1 

1.273 

5.09 

58 

364.  42 

10  568 

9.  23 

267.  70 

9 

56.  55 

254.5 

1.432 

6.45 

59 

370.  71 

10  936 

9.  39 

277.01 

10 

62.  83 

314.2 

1.592 

7.96 

60 

376. 99 

11310 

9.55 

286. 48 

11 

69. 12 

380. 1 

1.  751 

9.  63 

61 

383.  27 

11690 

9.71 

296. 11 

12 

75.40 

452.4 

1.910 

11.46 

62 

389.  56 

12  076 

9.  87 

305.  90 

13 

81.68 

530.9 

2. 069 

13.45 

63 

395.  84 

12  469 

10.03 

315.  84 

14 

87.96 

615.8 

2.  228 

15.60 

64 

402. 12 

12  868 

10. 19 

325. 9i 

15 

94.  25 

706.9 

2.  387 

17.  90 

65 

408.  41 

13  273 

10.35 

336.  21 

16 

100.  53 

804.2 

2.  546 

20.37 

66 

414.  69 

13  685 

10.  50 

346.64 

17 

106.  81 

907.9 

2.  706 

23.00 

67 

420.  97 

14  103 

10.  66 

357.  22 

18 

113. 10 

1 017.  9 

2.  865 

25.  78 

68 

427.  26 

14  527 

10.  82 

367.  97 

19 

119.  38 

1 134. 1 

3.  024 

28.  73 

69 

433.  54 

14  957 

10.98 

378.  87 

20 

125.  66 

1 256.  6 

3. 183 

31.83 

70 

439.  82 

15  394 

11.14 

389.  93 

21 

131.95 

1 385.  4 

3.342 

35.09 

71 

446. 11 

15  837 

11.30 

401. 15 

22 

138.  23 

1 520.  5 

3.  501 

38.52 

72 

452.39 

16  286 

11.46 

412.  53 

23 

144.  51 

1 661.  9 

3,661 

42.  10 

73 

458.  67 

16  742 

11.62 

424.  07 

24 

150.  80 

1 809.  6 

3.820 

45.84 

74 

464.  96 

17  203 

11.  78 

435.  77 

25 

157.  08 

1 963.  5 

3.  979 

49.  74 

75 

471.  24 

17  671 

11.94 

447.  62 

26 

163.  36 

2 123.  7 

4. 138 

53.  79 

76 

477.  52 

18  146 

12. 10 

459.64 

27 

169.  65 

2 290.  2 

4.  297 

58.  01 

77 

483.  81 

18  627 

12.  25 

471.  81 

28 

175. 93 

2 463. 0 

4.  456 

62.  39 

78 

490.09 

19113 

12.41 

484. 15 

29 

182.  21 

2 642. 1 

4. 615 

66.  92 

79 

496.  37 

19  607 

12.57 

496.64 

30 

188.  50 

2 827.  4 

4.  775 

71.62 

80 

502. 65 

20  106 

12.  73 

509. 30 

31 

194.  78 

3 019. 1 

4.934 

76.  47 

81 

508.  94 

20  612 

12.  89 

522. 11 

32 

201.  06 

3 217.0 

5.093 

81.49 

82 

515.22 

21 124 

13.05 

535.  08 

33 

207.  35 

3 421.  2 

5.252 

86.  66 

S3 

521.  50 

21642 

13.  21 

548.  21 

34 

213. 63 

3 631.  7 

5.411 

91.99 

84 

527.  79 

22  167 

13.37 

561.  50 

35 

219. 91 

3 848.  5 

5.570 

97.  48 

85 

534.  07 

22  698 

13.53 

574.  95 

36 

226. 19 

4 071.5 

5.730 

103.  13 

86 

540.  35 

23  235 

13.69 

588.  55 

37 

232. 48 

4 300.  8 

5.889 

108.  94 

87 

546. 64 

23  779 

13.  85 

602.  32 

38 

238.  76 

4 536.  5 

6.  048 

114.  91 

88 

552.  92 

24  328 

14.  01 

616.  25 

39 

245.  04 

4 778. 4 

6.  207 

121.  04 

89 

559.  20 

24SSS 

14. 16 

630. 33 

40 

251.33 

5 026.  5 

6.  366 

127.  32 

90 

565.  49 

25  447 

14.32 

644. 58 

41 

257. 61 

5 281.  0 

6.525 

133.  77 

91 

571.  77 

26  016 

14.  48 

658.  98 

42 

263.  89 

5 541.8 

6.685 

140.  37 

92 

578.  05 

26  590 

14.64 

673.  54 

43 

270. 18 

5 808.  8 

6.  844 

147. 14 

93 

584.  34 

27  172 

14.80 

688.  27 

44 

276. 46 

6 082. 1 

7.003 

154.  06 

94 

590.  62 

27  759 

14.96 

703. 15 

45 

282.  74 

6 361.7 

7.162 

161. 14 

95 

596. 90 

28  353 

15. 12 

718.  19 

46 

289.  03 

6 647.  6 

7.321 

168.  39 

96 

603. 19 

28  953 

15.28 

733. 39 

47 

295.  31 

6 939.  8 

7.  480 

175.  79 

97 

609.  47 

29  559 

15.44 

748.  74 

48 

301.  59 

7 238.  2 

7.  639 

183.  35 

98 

615.  75 

30172 

15.60 

764.  26 

49 

307.  88 

7 543.  0 

7.  799 

191.  07 

99 

622.04 

30  791 

15.  76 

779.94 

50 

314.  16 

7 854.  0 

7.  958 

198.  94 

100 

628.  32 

31416 

15.92 

795.  77 

n 

2lrJl 

JTO* 

1 

27t” 

47t 

n 

2nn 

1 

27 


TABLE  III 


FIVE-PLACE  MANTISSAS 
OF  THE  COMMON  LOGARITHMS  OF 
INTEGERS  FROM  1 TO  10,000 

On  this  page  the  logarithms  of  integers  from  1 to  100  are  given  in  full, 
with  characteristics  as  well  as  mantissas.  On  account  of  the  great  dif- 
ferences between  the  successive  mantissas,  interpolation  cannot  safely  be 
employed  on  this  page. 

In  the  remainder  of  the  table  only  the  mantissas  are  given. 

In  general,  an  error  of  one  unit  may  appear  in  the  last  figure  of  any 
interpolated  value. 

Table  III  is  to  be  used  when  accuracy  is  required  to  more  than  four 
figures  in  the  results.  In  general,  the  resrdts  will  be  accurate  to  five  figures. 


1-100 


N 

log 

N 

log 

N 

log 

N 

log 

X 

log 

1 

0.  00  000 

21 

1.  32  222 

41 

1.  61  278 

61 

1.  78  533 

81 

1.  90  849 

2 

0.  30  103 

22 

1.  34  242 

42 

1.  62  325 

62 

1.  79  239 

82 

1.  91  381 

3 

0.  47  712 

23 

1.  36  173 

43 

1.  63  347 

63 

1.  79  934 

S3 

1.  91  908 

4 

0.  60  2()6 

24 

1.  38  021 

44 

1.  64  345 

64 

1.80  618 

84 

1.92  428 

5 

0.  69  897 

25 

1.  39  794 

45 

1.  65  321 

65 

1.  81  291 

85 

1.  92  942 

6' 

0.  77  815 

26 

1.  41  497 

46 

1.  66  276 

66 

1.  81  954 

86 

1.  93  450 

7 

0.  84  510 

27 

d.  43  136 

47 

1.  67  210 

67 

1.  82  607 

87 

1.  93  952 

.8 

0.  90  309 

28 

1.  44  716 

48 

1.  68  124 

68 

1.  83  251 

88 

1.  94  448 

9 

0.  95  424 

29 

1.  46  240 

49 

1.  69  020 

69 

1.  83  885 

89 

1.  94  939 

10 

1.00  000 

30 

1.  47  712 

.50 

1.  69  897 

70 

1.  84  510 

90 

1.  95  424 

u 

11 

/ 

1.04  139 

31 

1.49136 

51 

1.  70  757 

71 

1.  85  126 

91 

1.  95  904 

12 

1.  07  918 

32 

1.  50  515 

52 

1.  71  600 

72 

1.  85  733 

92 

1.96  379 

13 

1. 11  394 

33 

1.  51851 

53 

1.  72  428 

73 

1.  86  332 

93 

1.  96  848 

14 

1.  14  613 

34 

1.  53  148 

54 

1.  73  239 

74 

1.  86  923 

94 

1.97  313 

IS 

1.  17  609^ 

35 

1.  54  407 

55 

1.  74  036 

75 

1.  87  506 

95 

1.  97  772 

16 

1.  20  412 

36 

1.  55  630 

56 

1.  74  819 

76 

1.  88  081 

96 

1.  98  227 

17 

1.  23  045 

37 

1.  56  820 

57 

1.  75  587 

77 

1.  88  649 

97 

1.  98  677 

18 

1.  25  527 

38 

1.  57  978 

58 

1.  76  343 

78 

1.  89  209 

98 

1.99123 

19 

1.27  875 

39 

1.  59  106 

59 

1.  77  OSS 

79 

1.  89  763 

99 

1.99  564 

20 

1.  30  103 

40 

1.  60  206 

60 

1.  77  815 

80 

1.  90  309 

100 

2.  00000 

N 

log 

X 

log 

N 

log 

X 

log 

X 

log 

1-100 


28 


100-150 


N 

O 

1 

2 

3 

4 

6 

6 

7 

8 

9 

100 

00  000 

00  043 

00  087 

00130 

00173 

00  217 

00  260 

00  303 

00  346 

00  389 

101 

432 

475 

518 

561 

604 

647 

689 

732 

775 

817 

102 

860 

903 

945 

988 

01  030 

01 072 

01  115 

01 157 

01 199 

01  242 

103 

01  284 

01  326 

01368 

01  410 

452 

494 

536 

578- 

620 

662 

104 

703 

745 

787 

828 

870 

912 

953 

995 

02  036 

02  078 

105 

02119 

02  160 

02  202 

02  243 

02  284 

02  325 

02  366 

02  407 

02  449 

02  490 

106 

531 

572 

612 

653 

694 

735 

776 

816 

857 

898 

107 

938 

979 

03  019 

03  060 

03  100 

03  141 

03  181 

03  222 

03  262 

03  302 

108 

03  342 

03  383 

423 

463 

503 

543 

583 

623 

663 

703 

109 

743 

782 

822 

862 

902 

941 

981 

04021 

04060 

04100 

no 

04  139 

04179 

04  218 

04  258 

04  297 

04  336 

04  376 

04  415 

04  454 

04  493 

111 

532 

571 

610 

650 

689 

727 

766 

805 

844 

883 

112 

922 

961 

999 

05  038 

05  077 

05  115 

05  15^ 

05  192 

05  231 

05  269 

113 

05  308 

05  346 

05  38i 

423 

461 

500 

538 

576 

614 

652 

114 

690 

729 

767 

80S 

843 

881 

918 

956 

994 

06  032 

115 

06  070 

06  108 

06  145 

06  183 

06  221 

06  258 

06  296 

06  333 

06  371 

06  408 

116 

446 

483 

521 

558 

595 

633 

670 

707 

744 

781 

117 

819 

856 

893 

930 

967 

07  004 

07  041 

07  078 

07115 

07  151 

118 

07  188 

07  225 

07  262 

07  298 

07  335 

372 

408 

445 

482 

518 

119 

555 

591 

628 

664 

700 

737 

773 

809 

846 

882 

120 

07  918 

07  954 

07  990 

08  027 

08  063 

08  099 

08  135 

08171 

08  207 

08  243 

121 

08  279 

08  314 

08  350 

386 

422 

458 

493 

529 

565 

600 

122 

636 

672 

707 

743 

778 

814 

849 

884 

920 

955 

123 

991 

09  026 

09  061 

09  096 

09132 

09167 

09  202 

09  237 

09  272 

09307 

124 

09  342 

377 

412 

447 

482 

517 

552 

587 

621 

656 

125 

09  691 

09  726 

09  760 

09  795 

09  830 

09  864 

09  899 

09  934 

09  968 

10  003 

126 

10  037 

10  072 

10  106 

10  140 

10175 

10  209 

10  243 

10  278 

10  312 

346 

127 

380 

415 

449 

483 

517 

551 

585 

619 

653 

687 

128 

721 

755 

789 

823 

857 

890 

924 

958 

992 

11  025 

129 

11059 

11093 

11 126 

11 160 

11  193 

11227 

11261 

11294 

11327 

361 

130 

11394 

11  428 

11461 

11494 

11  528 

11  561 

11594 

11628 

11661 

11694 

131 

727 

760 

793 

826 

860 

893 

926 

959 

992 

12  024 

132 

12  057 

12  090 

12  123 

12  156 

12  189 

12  222 

12  254 

12  287 

12  320 

352 

133 

385 

418 

450 

483 

516 

548 

581 

613 

646 

678 

134 

710 

743 

775 

808 

840 

872 

90S 

937 

969 

13  001 

135 

13  033 

13  066 

13  098 

13  130 

13  162 

13  194 

13  226 

13  258 

13  290 

13  322 

136 

354 

386 

418 

450 

481 

513 

545 

577 

609 

640 

137 

672 

704 

735 

767 

799 

830 

862 

893 

925 

956 

138 

988 

14  019 

14  051 

14  082 

14  114 

14  145 

14176 

14  208 

14  239 

14  270 

139 

14  301 

333 

364 

395 

426 

457 

489 

520 

551 

582 

140 

14  613 

14  644 

14  675 

14  706 

14  737 

14  768 

14  799 

14  829 

14  860 

14  891 

141 

922 

953 

983 

15  014 

IS  045 

15  076 

IS  106 

IS  137 

IS  168 

15  198 

142 

15  229 

15  259 

15  290 

320 

351 

381 

412 

442 

473 

503 

143 

534 

564 

594 

625 

655 

685 

715 

746 

776 

806 

144 

836 

866 

897 

927 

957 

987 

16  017 

16  047 

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145 

16  137 

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16  227 

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16  346 

16  376 

16406 

146 

435 

465 

495 

524 

554 

584 

613 

643 

673 

702 

147 

732 

761 

791 

820 

850 

879 

909 

938 

967 

997 

148 

17  026 

17  056 

17  085 

17114 

17  143 

17173 

17  202 

17  231 

17  260 

17  289 

149 

319 

348 

377 

406 

435 

464 

493 

522 

551 

580 

150 

17  609 

17  638 

17  667 

17  696 

17  725 

17  754 

17  782 

17  811 

17  840 

17  869 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100-150 


150-200 


29 


N 

O 

1 

2 

3 

4 

6 

6 

7 

8 

9 

160 

17  609 

17  638 

17  667 

17  696 

17  725 

17  754 

17  782 

17  811 

17  840 

17  869 

151 

898 

926 

955 

984 

18  013 

18  041 

18  070 

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152 

18  184 

18  213 

18  241 

18  270 

298 

327 

355 

384 

412 

441 

153 

469 

498 

526 

554 

583 

611 

639 

667 

696 

724 

154 

752 

780 

808 

837 

865 

893 

921 

949 

977 

19  005 

165 

19  033 

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19117 

19  145 

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19  229 

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156 

312 

340 

368 

396 

424 

451 

479 

507 

535 

562 

157 

590 

618 

645 

673 

700 

728 

756 

783 

811 

838 

158 

866 

893 

921 

948 

976 

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20112 

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20140 

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276 

303 

330 

358 

385 

160 

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683 

710 

737 

763 

790 

817 

844 

871 

898 

925' 

162 

952 

978 

21005 

21032 

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21085 

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21192 

163 

21  219 

21245 

272 

299 

325 

352 

378 

405 

431 

458 

164 

484 

511 

537 

564 

590 

617 

643 

669 

696 

722 

165 

21  748 

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21801 

21  827 

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21932 

21958 

21985 

166 

22  011 

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272 

298 

324 

350 

376 

401 

427 

453 

479 

505 

168 

531 

557 

583 

608 

634 

660 

686 

712 

737 

763 

169 

789 

814 

840 

866 

891 

917 

943 

968 

994 

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170 

23  045 

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23  274 

171 

300 

325 

350 

376 

401 

426 

452 

477 

502 

528 

172 

553 

/ 578 

603 

629 

654 

679 

704 

729 

754 

779 

173 

805, 

830 

85i 

880 

905 

930 

955 

980 

24  005 

24030 

174 

24  055 

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24  130 

2415i 

24180 

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24  229 

254 

279 

176 

24  304 

24  329 

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176 

551 

576 

601 

625 

650 

674 

699 

724 

748 

773 

177 

797 

822 

846 

871 

895 

920 

944 

969 

993 

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178 

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179 

285 

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334 

358 

382 

406 

431 

455 

479 

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180 

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181 

768 

792 

816 

840 

864 

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912 

935 

959 

983 

182 

26007 

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183 

245 

269 

293 

316 

340 

364 

387 

411' 

435 

458 

184 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 

185 

26  717 

26  741 

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26928 

186 

951 

975 

998 

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27114 

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187 

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277 

300 

323 

346 

370 

393 

188 

416 

439 

462 

485 

508 

531 

554 

577 

600 

623 

189 

646 

669 

692 

715 

738 

761 

784 

807 

830 

852 

190 

27  875 

27898 

27  921 

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191 

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240 

262 

285 

307 

192 

330 

353 

375 

398 

421 

443 

466 

488 

511 

533 

193 

556 

578 

601 

623 

646 

668 

691 

713 

735 

758 

194 

780 

803 

825 

847 

870 

892 

914 

937 

959 

981 

195 

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29  026 

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196 

226 

248 

270 

292 

314 

336 

358 

380 

403 

425 

197 

447 

469 

491 

513 

535 

557 

579 

601 

623 

645 

198 

667 

688 

710 

732 

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798 

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842 

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199 

885 

907 

929 

951 

973 

994 

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30  060 

30  081 

200 

30  103 

30  125 

30  146 

30  168 

30  190 

30  211 

30  233 

30  255 

30  276 

30  298 

N 

O 

1 

2 

3 

4 

6 

6 

7 

8 

9 

150-200 


30 


200-250 


N 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

200 

30  103 

30125 

30  146 

30  168 

30  190 

30  211 

30  233 

30  255 

30  276 

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201 

320 

341 

363 

384 

406 

428 

449 

471 

492 

514 

202 

535 

557 

578 

600 

621 

643, 

664 

685 

707 

728 

203 

750 

771 

792 

814 

835 

856 

878 

899 

920 

942 

204 

963 

984 

31006 

31027 

31048 

31069 

31  091 

31 112 

31  133 

31 154 

206 

31  175 

31  197.> 

31218 

31  239 

31260 

31281 

31  302 

31323 

31345 

31366 

206 

387 

40^ 

429 

450 

471 

492 

513 

534 

555 

576 

207 

597 

618 

639 

660 

681 

702 

723 

744 

765 

785 

208 

806 

827 

848 

869 

890 

911 

931 

952 

973 

994 

209 

32  015 

32  035 

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210 

32  222 

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211 

428 

449 

469 

490 

510 

531 

552 

572 

593 

613 

212 

634 

654 

675 

695 

715 

736 

756 

777 

797 

818 

213 

838 

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879 

899 

919 

940 

960 

980 

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33  021 

214 

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203 

224 

216 

33  244 

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33  405 

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445 

465 

486 

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526 

546 

566 

586 

606 

626 

217 

646 

666 

686 

706 

726 

746 

766 

786 

806 

826 

218 

846 

866 

885 

905 

925 

945 

965 

985 

34  005 

34  025 

219 

34  044 

34  064 

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203 

223 

220 

34  242 

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34400 

34  420 

221 

439 

459 

479 

498 

518 

537 

557 

577 

596 

616 

222 

635 

655 

674 

694 

713 

733 

753 

772 

792 

811 

223 

830 

850 

869 

889 

908 

928 

947 

967 

986 

35  005 

224 

35  025 

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199 

226 

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411 

430 

449 

468 

488 

507 

526 

545 

564 

583 

227 

603 

622 

641 

660 

679 

698 

717 

736 

755 

774 

228 

793 

813 

832 

851 

870 

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908 

927 

946 

965 

229 

984 

36  003 

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36116 

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230 

36173 

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361 

380 

399 

418 

436 

455 

474 

493 

511 

530 

232 

549 

568 

586 

60S 

624 

642 

661 

680 

698 

717 

233 

736 

754 

773 

791 

810 

829 

847 

866 

884 

903 

234 

922 

940 

959 

< 977 

996 

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37107 

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291 

310 

328 

346 

365 

383 

401 

420 

438 

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237 

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511 

530 

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585 

603 

621 

639 

238 

658 

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712 

731 

749 

767 

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822 

239 

840 

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931 

949 

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202 

220 

238 

256 

274 

292 

310 

328 

346 

364 

242 

382 

399 

417 

435 

453 

471 

489 

507 

52i 

543 

243 

561 

578 

596 

614 

632 

650 

668 

686 

703 

721 

244 

739 

757 

775 

792 

810 

828 

846 

863 

. 881 

899 

246 

38  917 

38  934 

38  952 

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38987 

39  005 

39023 

39  041 

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246 

39  094 

39111 

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182 

199 

217 

235 

252 

247 

270 

287 

305 

322 

340 

358 

375 

393 

410 

428 

248 

445 

463 

480 

498 

515 

533 

550 

568 

585 

602 

249 

620 

637 

655 

672 

690 

707 

724 

742 

759 

777 

260 

39  794 

39  811 

39  829 

39  846 

39  863 

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39  898 

39  915 

39  933 

391950 

N 

O 

1 

2 

3 

4 

6 

6 

7 

8 

9 

200-250 


250-300 


31 


N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

250 

39  79+ 

39  811 

39  829 

39  S+6 

39  863 

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967 

985 

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252 

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175 

192 

209 

226 

243 

261 

278 

295  ^ 

253 

312 

329 

3+6 

36+ 

381 

398 

415 

432 

449 

466 

25+ 

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500 

. 518 

535 

552 

569 

586 

603 

620 

637 

255 

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82+ 

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926 

943 

960 

976 

257 

993 

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41027 

410+4 

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25S 

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179 

196 

212 

229 

246 

263 

280 

296 

313 

259 

330 

3+7 

363 

380 

397 

414 

430 

4+7 

464 

481 

260 

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66+ 

681 

697 

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731 

747 

764 

780 

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814 

262 

830 

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863 

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913 

929 

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963 

979 

263 

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26+ 

42  160 

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193 

210 

226 

2+3 

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275 

292 

308 

265 

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42  37+ 

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266 

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50+ 

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537 

553 

570 

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602 

619 

635 

267 

651 

667 

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700 

716 

732 

749 

765 

781 

797 

26S 

813 

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862 

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927 

943 

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297 

313 

329 

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377 

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457 

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521 

537 

553 

569 

584 

600 

273 

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632 

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680 

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712 

727 

743 

759 

27+ 

775 

791 

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823 

838 

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917 

275 

43  933 

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277 

2+8 

26+ 

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295 

311 

326 

342 

358 

373 

389 

278 

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436 

451 

467 

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529 

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560 

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623 

638 

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669 

685 

700 

280 

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4+  731 

4+7+7 

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871 

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917 

932 

948 

963 

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994 

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283 

179 

19+ 

209 

225 

240 

255 

271 

286 

301 

317 

28+ 

332 

3+7 

362 

378 

393 

408 

423 

439 

454 

469 

285 

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45  500 

45  515 

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45  591 

45  606 

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60  206 

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350-400 


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68  006 

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126 

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500 

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70  001 

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70  018 

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502 

70  070 

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088 

096 

105 

114  122  131  140  148 

503 

157 

165 

174 

183 

191 

200  209  217  226  234 

504 

243 

252 

260 

269 

278 

286  295  303  312  321 

605 

70  329 

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70  346 

70  355 

70  364 

70  372  70  381  70  389  70  398  70  406 

506 

415 

424 

432 

441 

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458  467  475  484  492 

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518 

526 

535 

544  552  561  569  578 

508 

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595 

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612 

621 

629  638  646  655  663 

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680 

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714  723  731  740  749 

610 

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70  800  70  808  70  817  70  825  70  834 

511 

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512 

927 

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961 

969  978  986  995  71003. 

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105 

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139  147  155  164  172 

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71  181 

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71  206 

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516 

265 

273 

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290 

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307  315  324  332  341 

517 

349 

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391  399  408  416  425 

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620 

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725  734  742  750  759 

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800 

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523 

850 

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867 

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892  900  90S  917  925 

524 

933 

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966 

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625 

72  016 

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526 

099 

107 

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140  148  156  165  173 

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181 

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713  722  730  738  746 

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72  835 

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538 

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119  127  135  143  151 

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640 

73  239 

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360  368  376  384  392 

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440  448  456  464  472 

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547 

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548 

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918  926  933  941  949 

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500-550 


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37 


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1 

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550 

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551 

115 

123 

131 

139 

147 

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162 

170 

178 

186 

552 

194 

202 

210 

218 

225 

233 

241 

249 

257 

265 

553 

273 

280 

288 

296 

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312 

320 

327 

335 

343 

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351 

359 

367 

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382 

390 

398 

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414 

421 

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74  429 

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539 

547 

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562 

570 

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586 

593 

601 

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617 

624 

632 

640 

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671 

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780 

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74  819 

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561 

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927 

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966 

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974 

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75  005 

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563 

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082 

089 

097 

105 

113 

120 

564 

128 

136 

143 

151 

159 

166 

174 

182 

189 

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565 

75  205 

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75  259 

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282 

289 

297 

305 

312 

320 

328 

335 

343 

351 

567 

358 

366 

374 

381 

389 

397 

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412 

420 

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542 

549 

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580 

670 

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740 

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080 

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103 

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118 

125 

133 

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163 

170 

178 

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193 

200 

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215 

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238 

245 

253 

260 

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275 

283 

290 

298 

305 

313 

320 

328 

335 

680 

76  343 

76  350 

76  358 

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76  373 

76  380 

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418 

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433 

440 

448 

455 

462 

470 

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500 

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530 

537 

545 

552 

559 

583 

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589 

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612 

619 

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634 

584 

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664 

671 

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686 

693 

701 

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76  716 

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159 

166 

173 

181 

188 

195 

203 

210 

217 

225 

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232 

240 

247 

254 

262 

269 

276 

283 

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298 

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305 

313 

320 

327 

335 

342 

349 

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364 

371 

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379 

386 

393 

401 

408 

415 

422 

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695 

77  452 

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525 

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539 

546 

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634 

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598 

670 

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714 

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801 

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77  815 

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N 

0 

1 

2 

3 

4 

5 

6 

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550-600 


38 


600-650 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

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600 

77  815 

77  822 

77  830 

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77  844 

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601 

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916 

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931 

938 

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960 

967 

974 

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78  003 

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082 

089 

097 

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104 

111 

118 

125 

132 

140 

147 

154 

161 

168 

605 

78  176 

78  183 

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606 

247 

254 

262 

269 

276 

283 

290 

297 

305 

312 

607 

319 

326 

333 

340 

347 

355 

362 

369 

376 

383 

608 

390 

398 

405 

412 

419 

426 

433 

440 

447 

455 

609 

462 

469 

476 

483 

490 

497 

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512 

519 

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610 

78  533 

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611 

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618 

625 

633 

640 

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661 

668 

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718 

725 

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615 

78  888 

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79  000 

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106 

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141 

148 

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162 

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183 

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309 

316 

323 

330 

337 

344 

351 

358 

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379 

386 

393 

400 

407 

414 

421 

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435 

442 

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470 

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560 

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657 

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713 

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734 

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79  934 

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140 

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80  277 

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265 

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81  291 

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81  325 

81  331 

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81  351 

O 

1 

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4 

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6 

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600-650 


650-700 


89 


N 

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1 

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770 

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480 

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536 

542 

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570 

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734 

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773 

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154 

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265 

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393 

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542 

548 

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570 

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135 

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146 

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162 

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173 

179 

184 

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200 

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227 

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238 

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255 

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266 

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276 

282 

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90  309 

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634 

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650 

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666 

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714 

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730 

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773 

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734 

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809 

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800 

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811 

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822 

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832 

838 

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810 

90  849 

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90  859 

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92  942 

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1 

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7 

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800-850 


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313 

318 

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349 

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551 

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866 

752 

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852 

857 

862 

'867 

872 

877 

882 

887 

892 

897 

869 

902 

907 

912 

917 

922 

927 

932 

937 

942 

947 

870 

93  952 

93  957 

93  962 

93  967 

93  972 

93  977 

93  982 

93  987 

93  992 

93  997 

871 

94  002 

94  007 

94  012 

94  017 

94  022 

94  027 

94  032 

94  037 

94  042 

94  047 

872 

052 

057 

062 

067 

072 

077 

082 

086 

091 

096 

873 

101 

106 

111 

116 

121 

126 

131 

136 

141 

146 

874 

151 

156 

161 

166 

171 

176 

181 

186 

191 

196 

875 

94  201 

94  206 

94  211 

94  216 

94  221 

94  226 

94  231 

94  236 

94  240 

94  245 

876 

250 

255 

260 

265 

270 

275 

280 

285 

290 

295 

877 

300 

305 

310 

315 

320 

325 

330 

335 

340 

345 

878 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

879 

399 

404 

409 

414 

419 

424 

429 

433 

438 

443 

880 

94  448 

94  453 

94  458 

94  463 

94  468 

94  473 

94  478 

94  483 

94  488 

94  493 

881 

498 

503 

507 

512 

517 

522 

527 

532 

537 

542 

882 

547 

552 

557 

562 

567 

571 

576 

581 

586 

591 

883 

596 

601 

606 

611 

616 

621 

626 

630 

635 

640 

884 

645 

650 

655 

660 

665 

670 

675 

680 

685 

689 

885 

94  694 

94  699 

94  704 

94  709 

94  714 

94  719 

94  724 

94  729 

94  734 

94  738 

886 

743 

748 

753 

758 

763 

768 

773 

778 

783 

787 

887 

792 

797 

802 

807 

812 

817 

822 

827 

832 

836 

888 

841 

846 

851 

856 

861 

866 

871 

876 

880 

885 

889 

890 

895 

900 

905 

^ 910 

915 

919 

924 

929 

934 

890 

94  939 

94  944 

94  949 

94  954 

94  959 

94  963 

94  968 

94  973 

94  978 

94  983 

891 

988 

993 

998 

95  002 

95  007 

95  012 

95  017 

95  022 

95  027 

95  032 

892 

95  036 

95  041 

95  046 

051 

056 

061 

066 

071 

075 

080 

893 

085 

090 

095 

100 

105 

109 

114 

119 

124 

129 

894 

134 

139 

143 

148 

153 

158 

163 

168 

173 

177 

895 

95  182 

95  187 

95  192 

95  197 

95  202 

95  207 

95  211 

95  216 

95  221 

95  226 

896 

231 

236 

240 

245 

250 

255 

260 

265 

270 

274 

897 

279 

284 

289 

294 

299 

303 

308 

313 

318 

323 

898 

328 

332 

337 

342 

347 

352 

357 

361 

366 

371 

899 

376 

381 

386 

390 

395 

400 

405 

410 

415 

419 

900 

95  424 

95  429 

95  434 

95  439 

95  444 

95  448 

95  453 

95  458 

95  463 

95  468 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

850-900 


44 


900-950 


N 

O 

1 

2 

3 

4 

6 

6 

7 

8 

9 

900 

95  424 

95  429 

95  434 

95  439 

95  444 

95  448 

95  453 

95  458 

95  463 

95  468 

901 

472 

477 

482 

487 

492 

497 

501 

506 

511 

516 

902 

521 

525 

530 

535 

540 

545 

550 

554 

559 

564 

903 

569 

574 

578 

583 

588 

593 

598 

602 

607 

612 

904 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 

905 

95  665 

95  670 

95  674 

95  679 

95  684 

95  689 

95  694 

95  698 

95  703 

95  708 

906 

713 

718 

722 

727 

732 

737 

742 

746 

751 

756 

907 

761 

766 

770 

775 

780 

785 

789 

794 

799 

804 

908 

809 

813 

818 

823 

828 

832 

837 

842 

847 

852 

909 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 

910 

95  904 

95  909 

95  914 

95  918 

95  923 

95  928 

95  933 

95  938 

95  942 

95  947 

911 

952 

957 

961 

966 

971 

976 

980 

985 

990 

995 

912 

999 

96  004 

96  009 

96  014 

96  019 

96  023 

96  028 

96  033 

96  038 

96  042 

913 

96  047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

914 

095 

099 

104 

109 

114 

118 

123 

128 

133 

137 

915 

96  142 

96  147 

96152 

96156 

96  161 

96166 

96171 

96175 

96  180 

96185 

916 

190 

194 

199 

204 

209 

213 

218 

223 

227 

232 

917 

237 

242 

246 

251 

256 

261 

265 

270 

275 

280 

918 

284 

289 

294 

298 

303 

308 

313 

317 

322 

327 

919 

332 

336 

341 

346 

350 

355 

360 

365 

369 

374 

920 

96  379 

96  384 

96  388 

96  393 

96  398 

96  402 

96  407 

96  412 

96  417 

96  421 

921 

426 

431 

435 

440 

445 

450 

454 

459 

464 

468 

922 

473 

478 

483 

487 

492 

497 

501 

506 

511 

515 

923 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 

924 

567 

572 

577 

581 

586 

591 

595 

600 

605 

609 

925 

96  614 

96  619 

96  624 

96  628 

96  633 

96  638 

96  642 

96  647 

96  652 

96  656 

926 

661 

666 

670 

675 

680 

685 

689 

694 

699 

703 

927 

70S 

713 

717 

722 

727 

731 

736 

741 

745 

750 

928 

755 

759 

764 

769 

774 

778 

783 

788 

792 

797 

929 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 

930 

96  848 

96  853 

96  858 

96  862 

96  867 

96  872 

96  876 

96  881 

96  886 

96  890 

931 

895 

900 

904 

909 

914 

918 

923 

928 

932 

937 

932 

942 

946 

951 

956 

960 

965 

970 

974 

979 

984 

933 

988 

993 

997 

97  002 

97  007 

97  011 

97  016 

97  021 

97  025 

97  030 

934 

97  035 

97  039 

97  044 

049 

053 

058 

063 

067 

072 

077 

935 

97  081 

97  086 

97  090 

97  095 

97  100 

97  104 

97  109 

97114 

97118 

97  123 

936 

128 

132 

137 

142 

146 

151 

155 

160 

165 

169 

937 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

938 

220 

225 

230 

234 

239 

243 

248 

253 

257 

262 

939 

267 

271 

276 

280 

285 

290 

294 

299 

304 

308 

940 

97  313 

97  317 

97  322 

97  327 

97  331 

97  336 

97  340 

97  345 

97  350 

97  354 

941 

359 

364 

368 

373 

377 

382 

387 

391 

396 

400 

942 

405 

410 

414 

419 

424 

428 

433 

437 

442 

447 

943 

451 

456 

460 

465 

470 

474 

479 

483 

488 

493 

944 

497 

502 

506 

511 

516 

520 

525 

529 

534 

539 

945 

97  543 

97  548 

97  552 

97  557 

97  562 

97  566 

97  571 

97  575 

97  580 

97  585 

946 

589 

594 

598 

603 

607 

612 

617 

621 

626 

630 

947 

635 

640 

644 

649 

653 

658 

663 

667 

672 

676 

948 

681 

685 

690 

695 

699 

704 

70S 

713 

717 

722 

949 

727 

731 

736 

740 

745 

749 

754 

759 

763 

768 

950 

97  772 

97  777 

97  782 

97  786 

97  791 

97  795 

97  800 

97  80+ 

97  809 

97  813 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

900-950 


950-1000 


45 


N 

o 

1 

2 

3 

4 

5 

6 

7 

8 

9 

950 

97  772 

97  777 

97  782 

97  786 

97  791 

97  795 

97  800 

97  804 

97  809 

97  813 

951 

818 

823 

827 

832 

836 

841 

845 

850 

855 

859 

952 

864 

868 

873 

877 

882 

886 

891 

896 

900 

905 

953 

909 

914 

918 

923 

928 

932 

937 

941 

946 

950 

954 

,955 

959 

964 

968 

973 

978 

982 

987 

991 

996 

965 

/ 98  000 

98  005 

98  009 

98  014 

98  019 

98  023 

98  028 

98  032 

98  037 

98  041 

956 

046 

050 

055 

059 

064 

068 

073 

078 

082 

087 

957 

091 

096 

100 

105 

109 

114 

118 

123 

127 

132 

958 

137 

141 

146 

150 

155 

159 

164 

168 

173 

177 

959 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 

960 

98  227 

98  232 

98  236 

98  241 

98  245 

98  250 

98  254 

98  259 

98  263 

98  268 

961 

272 

277 

281 

286 

290 

295 

299 

304 

308 

313 

962 

318 

322 

327 

331 

336 

340 

345 

349 

354 

358 

963 

363 

367 

372 

376 

381 

385 

390 

394 

399 

403 

964 

408 

412 

417 

421 

426 

430 

435 

439 

444 

448 

965 

98  453 

98  457 

98  462 

98  466 

98  471 

98  475 

98  480 

98  484 

98  489 

98  493 

966 

498 

502 

507 

511 

516 

520 

525 

529 

534 

538 

967 

543 

547 

552 

556 

561 

565 

570 

574 

579 

583 

968 

588 

592 

597 

601 

605 

610 

614 

619 

623 

628 

969 

632 

637 

641 

646 

650 

655 

659 

664 

668 

673 

970 

98  677 

98  682 

98  686 

98  691 

98  695 

98  700 

98  704 

98  709 

98  713 

98  717 

971 

722 

726 

731 

735 

740 

744 

749 

753 

758 

762 

972 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 

973 

811 

816 

820 

825 

829 

834 

838 

843 

847 

851 

974 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 

976 

98  900 

98  905 

98  909 

98  914 

98  918 

98  923 

98  927 

98  932 

98  936 

98  941 

976 

945 

949 

954 

958 

963 

967 

972 

976 

981 

985 

977 

989 

994 

998 

99  003 

99  007 

99  012 

99  016 

99  021 

99  025 

99  029 

978 

99  034 

99  038 

99  043 

047 

052 

056 

061 

065 

069 

074 

979 

078 

083 

087 

092 

096 

100 

105 

109 

114 

118 

980 

99123 

99  127 

99  131 

99136 

99  140 

99  145 

99149 

99154 

99158 

99  162 

981 

167 

171 

176 

180 

185 

189 

193 

198 

202 

207 

982 

211 

216 

220 

224 

229 

233 

238 

242 

247 

251 

983 

255 

260 

264 

269 

273 

277 

282. 

286 

291 

295 

984 

300 

304 

308 

313 

317 

322 

326 

330 

335 

339 

985 

99  344 

99  348 

99  352 

99  357 

99  361 

99  366 

99  370 

99  374 

99  379 

99  383 

986 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

987 

432 

436 

441 

445 

449 

454 

458 

463 

467 

471 

988 

476 

480 

484 

489 

493 

498 

502 

506 

511 

515 

989 

520^ 

524 

528 

533 

537 

542 

546 

550 

555 

559 

990 

99  564 

99  568 

99  572 

99  577 

99  581 

99  585 

99  590 

99  594 

99  599 

99  603 

991 

607 

612 

616 

621 

625 

629 

634 

638 

642 

647 

992 

651 

656 

660 

664 

669 

673 

677 

682 

686 

691 

993 

695 

699 

704 

708 

712 

717 

721 

726 

730 

734 

994 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 

996 

99  782 

99  787 

99  791 

99  795 

99  800 

99  804 

99  808 

99  813 

99  817 

99  822 

996 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 

997 

870-  874 

878 

883 

887 

891 

896 

900 

904 

909 

998 

913 

917 

922 

926 

930 

935 

939 

944 

948 

952 

999 

V 957 

961 

965 

970 

974 

978 

983 

987 

991 

996 

lOOO 

00  000 

00  004 

00  009 

00  013 

00  017 

00  022 

00  026 

00  030 

00  035 

00  039 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

950-1000 


46 


PKOPOETIONAL  PARTS  OF  DIFFERENCES 


47 


This  table  contains  the  proportional  parts  of  differences  from  1 to  100. 
For  example,  if  the  difference  between  two  numbers  is  73,  0.7  of  this 
difference  is  51.1. 

D 

1 

2 

3 

4 

5 

a 

7 

8 

9 

51 

5.1 

10.2 

15.3 

20.4 

25.5 

30.6 

35.7 

40.8 

45.9 

52 

5.2 

10.4 

15.6 

20.8 

26.0 

31.2 

36.4 

41.6 

46.8 

53 

5.3 

10.6 

15.9 

21.2 

26.5 

31.8 

37.1 

42.4 

47.7 

54 

5.4 

10.8 

16.2 

21.6 

27.0 

32.4 

37.8 

43.2 

48.6 

55 

5.5 

11.0 

16.5 

22.0 

27.5 

33.0 

38.5 

44.0 

49.5 

56 

5.6 

11.2 

16.8 

22.4 

28.0 

33.6 

39.2 

44.8 

50.4 

57 

5.7 

11.4 

17.1 

22.8 

28.5 

34.2 

39.9 

45.6 

51.3 

58 

5.8 

11.6 

17.4 

23.2 

29.0 

34.8 

40.6 

46.4 

52.2 

59 

5.9 

11.8 

17.7 

23.6 

29.5 

35.4 

41.3 

47.2 

53.1 

60 

6.0 

12.0 

18.0 

24.0 

30.0 

36.0 

42.0 

48.0 

54.0 

61 

6.1 

12.2 

18.3 

24.4 

30.5 

36.6 

42.7 

48.8 

54.9 

62 

6.2 

12.4 

18.6 

24.8 

31.0 

37.2 

43.4 

49.6 

55.8 

63 

6.3 

12.6 

18.9 

25.2 

31.5 

37.8 

44.1 

50.4 

56.7 

64 

6.4 

12.8 

19.2 

25.6 

32.0 

38.4 

44.8 

51.2 

57.6 

65 

6.5 

13.0 

19.5 

26.0 

32.5 

39.0 

45.5 

52.0 

58.5 

66 

6.6 

13.2 

19.8 

26.4 

33.0 

39.6 

46.2 

52.8 

59.4 

67 

6.7 

13.4 

20.1 

26.8 

33.5 

40.2 

46.9 

53.6 

60.3 

68 

6.8 

13.6 

20.4 

27.2 

34.0 

40.8 

47.6 

54.4 

61.2 

69 

6.9 

13.8 

20.7 

27.6 

34.5 

41.4 

48.3 

55.2 

62.1 

70 

7.0 

14.0 

21.0 

28.0 

35.0 

42.0 

49.0 

56.0 

63.0 

71 

7.1 

14.2 

21.3 

2S.4 

35.5 

42.6 

49.7 

56.8 

63.9 

72 

7.2 

14.4 

21.6 

28.8 

36.0 

43.2 

50.4 

57.6 

64.8 

73 

7.3 

14.6 

21.9 

29.2 

36.5 

43.8 

51.1 

58.4 

65.7 

74 

7.4 

14.8 

22.2 

29.6 

37.0 

44.4 

51.8 

59.2 

66.6 

75 

7.5 

15.0 

22.5 

30.0 

37.5 

45.0 

52.5 

60.0 

67.5 

76 

7.6 

15.2 

22.8 

30.4 

38.0 

45.6 

53.2 

60.8 

68.4 

77 

7.7 

15.4 

23.1 

30.8 

38.5 

46.2 

53.9 

61.6 

69.3 

78 

7.8 

15.6 

23.4 

31.2 

39.0 

46.8 

54.6 

62.4 

70.2 

79 

7.9 

15.8 

23.7 

31.6 

39.5 

47.4 

55.3 

63.2 

71.1 

80 

8.0 

16.0 

24.0 

32.0 

40.0 

48.0 

56.0 

64.0 

72.0 

81 

8.1 

16.2 

24.3 

32.4 

40.5 

48.6 

56.7 

64.8 

72.9 

82 

8.2 

16.4 

24.6 

32.8 

41.0 

49.2 

57.4 

65.6 

73.8 

83 

8.3 

16.6 

24.9 

33.2 

41.5 

49.8 

58.1 

66.4 

74.7 

84 

8.4 

16.8 

25.2 

33.6 

42.0 

50.4 

58.8 

67.2 

75.6 

85 

8.5 

17.0 

25.5 

34.0 

42.5 

51.0 

59.5 

68.0 

76.5 

86 

8.6 

17.2 

25.8 

34.4 

43.0 

51.6 

60.2 

68.8 

77.4 

87 

8.7 

17.4 

26.1 

34.8 

43.5 

52.2 

60.9 

69.6 

78.3 

88 

8.8 

17.6 

26.4 

35.2 

44.0 

52.8 

61.6 

70.4 

79.2 

89 

8.9 

17.8 

26.7 

35.6 

44.5 

53.4 

62.3 

71.2 

80.1 

90 

9.0 

18.0 

27.0 

36.0 

45.0 

54.0 

63.0 

72.0 

81.0 

91 

9.1 

18.2 

27.3 

36.4 

45.5 

54.6 

63.7 

72.8 

81.9 

92 

9.2 

18.4 

27.6 

36.8 

46.0 

55.2 

64.4 

73.6 

82.8 

93 

9.3 

18.6 

27.9 

37.2 

46.5 

55.8 

65.1 

74.4 

83.7 

94 

9.4 

18.8 

28.2 

37.6 

47.0 

56.4 

65.8 

75.2 

84.6 

95 

9.5 

19.0 

28.5 

38.0 

47.5 

57.0 

66.5 

76.0 

85.5 

96 

9.6 

19.2 

28.8 

38.4 

48.0 

57.6 

67.2 

76.8 

86.4 

97 

9.7 

19.4 

29.1 

38.8 

48.5 

58.2 

67.9 

77.6 

87.3 

98 

9.8 

19.6 

29.4 

39.2 

49.0 

58.8 

68.6 

78.4 

88.2 

99 

9.9 

19.8 

29.7 

39.6 

49.5 

59.4 

69.3 

79.2 

89.1 

100 

10.0 

20.0 

30.0 

40.0 

50.0 

60.0 

70.0 

80.0 

90.0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

48 


TABLE  V.  LOGARITHMS  OF  CONSTANTS 


Number 

Loo 

Number 

Log 

Circle  = 360° 

2.55630 

772  9.86960 

0.99430 

= 21,600' 

= 1,296,000" 

4.33445 

6.11261 

^ = 0.10132 

TT^ 

9.00570  - 

10 

IT  = 3.14159 

0.49715 

= 1.77245 

0.24857 

2 77  = 6.28319 
4 77  = 12.56637 

0.79818 

1.09921 

= 0.56419 

Vtt 

9.75143  - 

10 

4 77 

— = 4.18879 
3 

0.62209 

= 1.12838 

0.05246 

- = 0.78540 
4 

9.89509  - 

10 

= 1.46459 

0.16572 

- = 0.52360 
6 

9.71900  - 

10 

= 0.68278 

Vt7 

9.83428  - 

10 

- = 0.31831 

7T 

9.50285  - 

10 

= 0.62035 

\477 

9.79264  - 

10 

— = 0.15915 
2 7T 

9.20182  - 

10 

= 0.80600 

9.90633  - 

10 

V2  = 1.41421 

0.15052 

V2  = 1.25992 

0.10034 

Vs  = 1.73205 

0.23856 

Vs  = 1.44225 

0.15904 

V5  = 2.23606 

0.34949 

Vb  = 1.70997 

0.23299 

Ve  = 2.44948 

0.38908 

Ve  = 1.81712 

0.25938 

1 radian  = 

7T 

1°  = VIL  radians 
180 

= 57.2958° 

1.75812 

1°  = 0.01745  radians 

8.24188  - 

10 

= 3437.75' 

3.53627 

1'  = 0.00029  radians 

6.46373  - 

10 

= 206,264.81" 

5.31443 

1"  = 0.000005  radians 

4.68557  — 

10 

Base  of  natural  logs.,  e 

log.^0  e = log^o  2.71828 

0.43429 

e = 2.71828 

0.43429 

l:logjo  6 = 2.302585 

0.36222 

1 m.  = 39.3708  in. 

1.59517 

1 knot  = 6080.27  ft. 

3.78392 

= 1.0936  yd. 

0.03886 

= 1.1516  mi. 

0.06130 

= 3.2809  ft. 

0.51599 

1 lb.  At.  = 7000  gr. 

3.84510 

1 km.  = 0.6214  mi. 

9.79336  - 

10 

1 bu.  = 2150.42  cu.  in. 

3.33252 

1 mi.  = 1.6093  km. 

0.20664 

1 U.S.  gal.  = 231  cu.  in. 

2.36361 

1 oz.  Av.  = 28.3495  g. 

1.45254 

1 Brit.  gal.  = 277.463  cu.  in. 

2.44320 

11b.  Av.  = 453.5927  g. 

2.65666 

Earth’s  radii 

1 kg.  = 2.2046  lb. 

0.34333 

= 3963  mi. 

3.59802 

11.  = 1.0567  liq.  qt. 

0.02396 

and  3950  mi. 

3.59660 

1 liq.  qt.  = 0.9463  1. 

9.97603  - 

10 

1 ft. /lb.  = 0.1383  kg./m. 

9.14082  - 

10 

49 


TABLE  YI 

THE  LOGARITHMS 

OF  THE  TRIGONOMETRIC  FUNCTIONS 


From  0°  to  0°  3',  and  from  89°  57'  to  90°,  for  every  second 
From  0°  to  2°,  and  from  88°  to  90°,  for  every  ten  seconds 
From  1°  to  89°,  for  every  minute 


To  each  logarithm  — 10  is  to  be  appended 

log  sin  0° 


log  tan  = log  sin 
log  cos  = 10.00  000 


ff 

O' 

1' 

2' 

rr 

ff 

O' 

1' 

2' 

ff 

0 



6.  46  373 

6.  76  476 

60 

30 

6. 16  270 

6.  63  982 

6.  86  167 

30 

1 

4.68  557 

6.  47  090 

6.  76  836 

59 

31 

6. 17  694 

6.  64  462 

6.  86  455 

29 

2 

4. 98  660 

6. 47  797 

6.  77  193 

58 

32 

6. 19  072 

6.  64  936 

6.  86  742 

28 

3 

5.15  270 

6.  48  492 

6.  77  548 

57 

33 

6.  20  409 

6.  65  406 

6.  87  027 

27 

4 

5.  28  763 

6. 49  175 

6.  77  900 

56 

34 

6.  21  70S 

6.  65  870 

6.  87  310 

26 

5 

5.  38  454 

6.  49  849 

6.  78  248 

55 

35 

6.  22  964 

6.  66  330 

6.  87  591 

26 

6 

5.  46  373 

6.  50  512 

6.  78  595 

54 

36 

6.  24  188 

6.  66  785 

6.  87  870 

24 

7 

5.  53  067 

6.  51 165 

6.  78  938 

S3 

37 

6.  25  378 

6.  67  235 

6.  88  147 

23 

8 

5.  58  866 

6.  51  808 

6.  79  278 

52 

38 

6.  26  536 

6.  67  680 

6.  88  423 

22 

9 

5. 63  982 

6.  52  442 

6.  79  616 

51 

39 

6.  27  664 

6.  68  121 

6.  88  697 

21 

10 

5.68  557 

6.  53  067 

6.  79  952 

50 

40 

6.  28  763 

6.  68  557 

6.  88  969 

20 

11 

5.  72  697 

6.  53  683 

6.  80  285 

49 

41 

6.  29  836 

6.  68  990 

6.  89  240 

19 

12 

5.  76  476 

6.  54  291 

6.  80  615 

48 

42 

6.  30  882 

6.  69  418 

6.  89  509 

18 

13 

5.  79  952 

6.  54  890 

6.  80  943 

47 

43 

6.31904 

6.  69  841 

6.  89  776 

17 

14 

5.  83  170 

6.  55  481 

6.  81  268 

46 

44 

6.  32  903 

6.  70  261 

6.  90  042 

16 

15 

5.  86  167 

6.  56  064 

6.  81  591 

45 

45 

6.  33  879 

6.  70  676 

6.  90  306 

16 

16 

5.88  969 

6.  56  639 

6.81911 

44 

46 

6.  34  833 

6.  71  088 

6.  90  568 

14 

17 

5. 91  602 

6.  57  207 

6.  82  230 

43 

47 

6.  35  767 

6.  71  496 

6.  90  829 

13 

18 

5.  94  OSS 

6.  57  767 

6.  82  545 

42 

48 

6.  36  682 

6.  71  900 

6.  91  088 

12 

19 

5.  96  433 

6.  58  320 

6.  82  859 

41 

49 

6.  37  577 

6.  72  300 

6.  91  346 

11 

20 

5.  98  660 

6.  58  866 

6.  83  170 

40 

50 

6.  38  454 

6.  72  697 

6.  91  602 

10 

21 

6.  00  779 

6.  59  406 

6.  83  479 

39 

51 

6.  39  315 

6.  73  090 

6.  91  857 

9 

22 

6.  02  800 

6.  59  939 

6.  83  786 

38 

52 

6.  40  158 

6.  73  479 

6.  92  110 

8 

23 

6.  04  730 

6.  60  465 

6.  84  091 

37 

53 

6.  40  985 

6.  73  865 

6.  92  362 

7 

24 

6. 06  579 

6.  60  985 

6.  84  394 

36 

54 

6.  41  797 

6.  74  248 

6.  92  612 

6 

25 

6.  08  351 

6.  61  499 

6.  84  694 

35 

55 

6.  42  594 

6.  74  627 

6.  92  861 

6 

26 

6. 10  OSS 

6.  62  007 

6.  84  993 

34 

56 

6.  43  376 

6.  75  003 

6.  93  109 

4 

27 

6. 11  694 

6.  62  509 

6.  85  289 

33 

57 

6. 44  145 

6.  75  376 

6.  93  355 

3 

28 

6. 13  273 

6.  63  006 

6.  85  584 

32 

58 

6.  44  900 

6.  75  746 

6.  93  599 

2 

29 

6. 14  797 

6.  63  496 

6.  85  876 

31 

59 

6.  45  643 

6.'76  112 

6.  93  843 

1 

30 

6. 16  270 

6.  63  982 

6.  86  167 

30 

60 

6.  46  373 

6.  76  476 

6.  94  085 

0 

tt 

59' 

58' 

57' 

ff 

ff 

59' 

68' 

67' 

ff 

log  cot  = log  cos 
log  sin=^  10.00  000 


89° 


log  cos 


50 


O' 


r ff 

log  sin 

log  cos 

log  tan 

t ft 

f ff 

log  sin 

log  cos 

log  tan 

f ff 

0 0 



10.00000 



60  0 

10  0 

7.  46  373 

10.00000 

7.46373 

50  0 

10 

5.68  557 

10.00000 

5.  68  557 

SO 

10 

7.  47  090 

10.00000 

7.  47  091 

SO 

20 

5.98  660 

10.00000 

5.98  660 

40 

20 

7.  47  797 

10.00000 

7.  47  797 

40 

30 

6. 16  270 

10.00000 

6. 16  270 

30 

30 

7.  48  491 

10.00000 

7.  48  492 

30 

40 

6.  28  763 

10.00000 

6.  28  763 

20 

40 

7.  49  175 

10.00000 

7. 49  176 

20 

50 

6.  38  454 

10.00000 

6.  38  454 

10 

SO 

7.  49  849 

10.00000 

7. 49  849 

10 

1 0 

6.  46  373 

10.00000 

6.  46  373 

59  0 

11  0 

7.  50  512 

10.00000 

7.  50  512 

49  0 

10 

6.  S3  067 

10.00000 

6.  53  067 

50 

10 

7.51  165 

10.00000 

7.  51 165 

50 

20 

6.  58  866 

10.00000 

6.  58  866 

40 

20 

7.  51  808 

10.00000 

7.51809 

40 

30 

6.  63  982 

10.00000 

6. 63  982 

30 

30 

7.  52  442 

10.00000 

7.  52  443 

30 

40 

6.  68  557 

10.00000 

6.  68  557 

20 

40 

7.  S3  067 

10.00000 

7.  53  067 

20 

SO 

6.  72  697 

10.00000 

6.  72  697 

10 

50 

7.  53  683 

10.00000 

7.  53  683 

10 

2 0 

6.  76  476 

10.00000 

6.  76  476 

58  0 

12  0 

7.  54  291 

10.00000 

7.  54  291 

48  0 

10 

6.  79  952 

10.00000 

6.  79  952 

50 

10 

7.  54  890 

10.00000 

7.  54  890 

50 

20 

6.  83  170 

10.00000 

6.  83  170 

40 

20 

7.  55  481 

10.00000 

7.  55  481 

40 

30 

6.  86  167 

10.00000 

6.  86  167 

30 

30 

7.  56  064 

10.00000 

7.  56  064 

30 

40 

6.  88  969 

10.00000 

6.  88  969 

20 

40 

7.  56  639 

10.00000 

7.  56  639 

20 

50 

6.  91  602 

10.00000 

6.  91  602 

10 

50 

7.  57  206 

10.00000 

7.  57  207 

10 

3 0 

6.  94  OSS 

10.00000 

6.  94  OSS 

57  0 

13  0 

7.  57  767 

10.00000 

7.  57  767 

47  0 

10 

6.  96  433 

10.00000 

6.  96  433 

50 

10 

7.  58  320 

10.00000 

7.  58  320 

50 

20 

6,  98  660 

10.00000 

6.  98  661 

40 

20 

7.  58  866 

10.00000 

7.  58  867 

40 

30 

7.  00  779 

10.00000 

7.00  779 

30 

30 

7.  59  406 

10.00000 

7.  59  406 

30 

40 

7.  02  800 

10.00000 

7.  02  800 

20 

40 

7.  59  939 

10.00000 

7.  59  939 

20 

SO 

7.  04  730 

10.00000 

7.  04  730 

10 

50 

7.  60  465 

10.00000 

7.  60  466 

10 

4 0 

7.  06  579 

10.00000 

7.  06  579 

56  0 

14  0 

7.  60985 

10.00000 

7.  60  986 

46  0 

10 

7.  08  351 

10.00000 

7.08  352 

50 

10 

7.  61  499 

10.00000 

7.  61  500 

50 

20 

7. 10  055 

10.00000 

7. 10  055 

40 

20 

7.  62  007 

10.00000 

7.  62  OOS 

40 

30 

7. 11  694 

10.00000 

7. 11  694 

30 

30 

7.  62  509 

10.00000 

7.  62  510 

30 

40 

7. 13  273 

10.00000 

7. 13  273 

20 

40 

7.  63  006 

10.00000 

7.  63  006 

20 

50 

7. 14  797 

10.00000 

7. 14  797 

10 

SO 

7.  63  496 

10.00000 

7.  63  497 

10 

5 0 

7. 16  270 

10.00000 

7. 16  270 

55  0 

15  0 

7.  63  982 

10.00000 

7.  63  982 

45  0 

10 

7.  17  694 

10.00000 

7. 17  694 

50 

10 

7.  64  461 

10.00000 

7.64  462 

50 

20 

7. 19  072 

10.00000 

7. 19  073 

40 

20 

7.  64  936 

10.00000 

7. 64  937 

40 

30 

7.  20  409 

10.00000 

7.  20  409 

30 

30 

7. 65  406 

10.00000 

7.  65  406 

30 

40 

7.  21  705 

10.00000 

7. 21  70S 

20 

40 

7.  65  870 

10.00000 

7.  65  87l 

20 

50 

7.  22  964 

10.00000 

7.  22  964 

10 

50 

7. 66  330 

10.00000 

7.  66  330 

10 

6 0 

7.  24  188 

10.00000 

7.  24  188 

54  0 

16  0 

7. 66  784 

10.00000 

7. 66  785 

440 

10 

7.  25  378 

10.00000 

7.  25  378 

50 

10 

7.  67  235 

10.00000 

7. 67  235 

50 

20 

7.  26  536 

10.00000 

7.  26  536 

40 

20 

7.  67  680 

10.00000 

7.  6/  680 

40 

30 

7.  27  664 

10.00000 

7.  27  664 

30 

30 

7. 68  121 

10.00000 

7.68121 

30 

40 

7.  28  763 

10.00000 

7.  28  764 

20 

40 

7.  68  557 

9.99999 

7.68  558 

20 

SO 

7.  29  836 

10.00000 

7.  29  836 

10 

50 

7.  68  989 

9.99999 

7.68  990 

10 

7 0 

7.  30  882 

10.00000 

7.  30  882 

53  0 

170 

7.69  417 

9.99  999 

7.  69  418 

43  0 

10 

7.  31  904 

10.00000 

7.  31  904 

50 

10 

7.  69  841 

9.  99  999 

7. 69  842 

50 

20 

7.  32  903 

10.00000 

7.  32  903 

40 

20 

7.  70  261 

9.  99  999 

7.  70  261 

40 

30 

7. 33  879 

10.00000 

7. 33  879 

30 

30 

7.  70  676 

9.  99  999 

7 . 70  t)/  / 

30 

40 

7.  34  833 

10.00000 

7. 34  833 

20 

40 

7.  71  OSS 

9. 99  999 

7.  71  OSS 

20 

50 

7.  35  767 

10.00000 

7. 35  767 

10 

50 

7.  71  496 

9.  99  999 

7.  71 496 

10 

8 0 

7.  36  682 

10.00000 

7.  36  682 

52  0 

18  0 

7.  71  900 

9.  99  999 

7.  71  900 

42  0 

10 

7.37  577 

10.00000 

7.37  577 

SO 

10 

7.  72  300 

9.  99  999 

7.  72  301 

50 

20 

7.  38  454 

10.00000 

7.  38  455 

40 

20 

7.  72  697 

9.  99  999 

7.  72  697 

40 

30 

7.  39  314 

10.00000 

7.39  315 

30 

30 

7.  73  090 

9. 99  999 

7.  73  090 

30 

40 

7.  40  158 

10.00000 

7.  40  158 

20 

40 

7.  73  479 

9. 99  999 

7.  73  480 

20 

50 

7. 40  985 

10.00000 

7.  40  985 

10 

50 

7.  73  865 

9. 99  999 

7.  73  866 

10 

9 0 

7.  41  797 

10.00000 

7.  41  797 

51  0 

190 

7.  74  248 

9. 99  999 

7.  74  248 

41  0 

10 

7.  42  594 

10.00000 

7. 42  594 

SO 

10 

7.  74  627 

9. 99  999 

7.  74  628 

50 

20 

7.  43  376 

10.00000 

7.43  376 

40 

20 

7.  75  003 

9.99  999 

7.  75  004 

40 

30 

7. 44  145 

10.00000 

7.  44  145 

30 

30 

7.  75  376 

9. 99  999 

V . / ^ 0 / / 

30 

40 

7.  44  900 

10.00000 

7. 44  900 

20 

40 

7-  75  745 

9.99  999 

7.  7^  746 

20 

SO 

7. 45  643 

10.00000 

7.  45  643 

10 

50 

7.  76  112 

9.  99  999 

7.  76  113 

10 

lOO 

7.  46  373 

10.00000 

7.  46  373 

50  0 

20  0 

7.  76475 

9.  99  999 

7.  76  476 

40  0 

log  cos 

log  sin 

log  cot 

/ ff 

f ff 

log  cos 

log  sin 

log  cot 

! ft 

89** 


O' 


51 


f ff 

log  sin 

log  cos 

log  tan 

f ff 

t ff 

log  sin 

log  cos 

log  tan 

/ ff 

200 

7.  76  475 

9. 99  999 

7.  76  476 

40  0 

30  0 

7.  94  084 

9.  99  998 

7. 94  086 

30  0 

10 

7.  76  836 

9.  99  999 

7.  76  837 

50 

10 

7.  94  325 

9. 99  998 

7. 94  326 

50 

20 

7.  77  193 

9.  99  999 

7.  77  194 

40 

20 

7.  94  564 

9.  99  998 

7.  94  566 

40 

30 

7.  77  548 

9.  99  999 

7.  77  549 

30 

30 

7. 94  802 

9. 99  998 

7.  94  804 

30 

40 

7.  77  899 

9.  99  999 

7.  77  900 

20 

40 

7.  95  039 

9.  99  998 

7.  95  040 

20 

50 

7.  78  248 

9.  99  999 

7.  78  249 

10 

50 

7.  95  274 

9.  99  998 

7.  95  276 

10 

210 

7.  78  594 

9.  99  999 

7.  78  595 

39  0 

31  0 

7. 95  508 

9.  99  998 

7.95  510 

29  0 

10 

7.  78  938 

9.  99  999 

7.  78  938 

50 

10 

7. 95  741 

9. 99  998 

7. 95  743 

50 

20 

7.  79  278 

9.  99  999 

7.  79  279 

40 

20 

7.  95  973 

9.  99  998 

7. 95  974 

40 

30 

7.  79  616 

9.  99  999 

7.79  617 

30 

30 

7.  96  203 

9.  99  998 

7. 96  205 

30 

40 

7.  79  952 

9.  99  999 

7.  79  952 

20 

40 

7.  96  432 

9. 99  998 

7.  96  434 

20 

50 

7.  80  284 

9. 99  999 

7.  80  285 

10 

50 

7.  96  660 

9.  99  998 

7.  96  662 

10 

220 

7.  80  615 

9. 99  999 

7.  80  615 

38  0 

32  0 

7.  96  887 

9.  99  998 

7. 96  889 

28  0 

10 

7.  80  942 

9. 99  999 

7.  80  943 

50 

10 

7.  97  113 

9.  99  998 

7.  97  114 

50 

20 

7.  81  268 

9.  99  999 

7.  81  269 

40 

20 

7.  97  337 

9.  99  998 

7.  97  339 

40 

30 

7.  81  591 

9.  99  999 

7.  81  591 

30 

30 

7.  97  560 

9.  99  998 

7. 97  562 

30 

40 

7.  81  911 

9. 99  999 

7.  81  912 

20 

40 

7.  97  782 

9. 99  998 

7.  97  784 

20 

50 

7.  82  229 

9.  99  999 

7.  82  230 

10 

50 

7.  98  003 

9. 99  998 

7. 98  005 

10 

230 

7.  82  545 

9.  99  999 

7.  82  546 

37  0 

33  0 

7.  98  223 

9.  99  998 

7.  98  225 

27  0 

10 

7.  82  859 

9.  99  999 

7.  82  860 

50 

10 

7.  98  442 

9.  99  998 

7. 98  444 

50 

20 

7.  83  170 

9. 99  999 

7.  S3  171 

40 

20 

7.  98  660 

9. 99  998 

7.  98  662 

40 

30 

7.  83  479 

9. 99  999 

7.  S3  480 

30 

30 

7.  98  876 

9. 99  998 

7. 98  878 

30 

40 

7.  83  786 

9. 99  999 

7.  S3  787 

20 

40 

7.  99  092 

9.  99  998 

7.  99  094 

20 

50 

7. 84  091 

9. 99  999 

7.  84  092 

10 

50 

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624 

510 

490 

IS 

43 

311 

300 

95  012 

04  988 

17 

43 

147 

612 

531 

465 

17 

44 

326 

288 

037 

963 

16 

44 

161 

600 

560 

440 

16 

45 

82  340 

87  277 

95  062 

04  938 

16 

45 

83  174 

86  589 

96  586 

03  414 

15 

46 

354 

266 

088 

912 

14 

46 

188 

577 

611 

389 

14 

47 

368 

255 

113 

887 

13 

47 

202 

565 

636 

364 

13 

48 

382 

243 

139 

861 

12 

48 

215 

554 

662 

338 

12 

49 

396 

232 

164 

836 

11 

49 

229 

542 

687 

313 

11 

50 

82  410 

87  221 

95  190 

04  810 

lO 

50 

83  242 

86  530 

96  712 

03  288 

10 

51 

424 

209 

215 

785 

9 

51 

256 

518 

738 

262 

9 

52 

439 

198 

240 

760 

8 

52 

270 

507 

763 

237 

8 

53 

453 

187 

266 

734 

7 

53 

283 

495 

788 

212 

7 

54 

467 

175 

291 

709 

6 

54 

297 

483 

814 

186 

6 

55 

82  481 

87  164 

95  31^ 

04  683 

6 

55 

83  310 

86472 

96  839 

03161 

5 

56 

495 

153 

342: 

658 

4- 

56 

324 

460 

864 

136 

4 

57 

509 

141 

368 

632 

3 

57 

338 

448 

890 

no 

3 

58 

523 

130 

393 

607 

2 

58 

351 

436 

915 

085 

2 

59 

537 

119 

418 

582 

1 

59 

365 

421 

940 

060 

1 

60 

82  551 

9 

87  107 

9 

95  444 

9 

04  556 

lO 

O 

60 

83  378 

9 

86413 

9 

96  966 

9 

03  034 

lO 

0 

9 

log  oos 

log  3m 

log  cot 

log  tan 

f 

t 

log  003 

log  sin 

log  oot 

log  tan 

t 

48' 


47 


43" 


77 


44" 


f 

log  sin 
9 

log  003 

9 

log  tan 
9 

log  cot 

10 

f 

r 

log  sin 

9 

log  cos 

9 

log  tan 

9 

log  oot 

10 

t 

0 

83  378 

86  413 

96  966 

03  034 

60 

0 

84  177 

85  693 

98  484 

01516 

60 

1 

392 

401 

96  991 

03  009 

59 

1 

190 

681 

509 

491 

59 

2 

405 

389 

97  016 

02  984 

58 

2 

203 

669 

534 

466 

58 

3 

419 

377 

042 

958 

57 

3 

216 

657 

560 

440 

57 

4 

432 

366 

067 

933 

56 

4 

229 

645 

585 

415 

56 

6 

83  446 

86  354 

97  092 

02  908 

55 

5 

84  242 

85  632 

98  610 

01  390 

55 

6 

459 

342 

118 

882 

54 

6 

255 

620 

635 

365 

54 

7 

473 

330 

143 

857 

53 

7 

269 

608 

661 

339 

53 

8 

486 

318 

168 

832 

52 

8 

282 

596 

686 

314 

52 

9 

500 

306 

193 

807 

51 

9 

295 

583 

711 

289 

51 

10 

83  513 

86  295 

97  219 

02  781 

50 

10 

84  308 

85  571 

98  737 

01  263 

50 

11 

527, 

540 

283 

244 

756 

49 

11 

321 

559 

762 

238 

49 

12 

271 

269 

731 

48 

12 

334 

547 

787 

213 

48 

13 

554 

259 

295 

705 

47 

13 

347 

534 

812 

188 

47 

14 

567 

247 

320 

680 

46 

14 

360 

522 

838 

162 

46 

15 

83  581 

86  235 

97  345 

02  655 

45 

15 

84  373 

85  510 

98  863 

01 137 

45 

16 

594 

223 

371 

629 

44 

16 

385 

497 

888 

112 

44 

17 

608 

211 

396 

604 

43 

17 

398 

485 

913 

087 

43 

18 

621 

200 

421 

579 

42 

18 

411 

473 

939 

061 

42 

19 

634 

188 

447 

553 

41 

19 

424 

460 

964 

036 

41 

20 

83  648 

86176 

97  472 

02  528 

40 

20 

84  437 

85  448 

98  989 

01  on 

40 

21 

661, 

/ 164 

497 

503 

39 

21 

450 

436 

99  015 

00  985 

39 

22 

674 

152 

523 

477 

38 

22 

463 

423 

040 

960 

38 

23 

688 

140 

548 

452 

37 

23 

476 

411 

065 

935 

37 

24 

701 

128 

573 

427 

36 

24 

489 

399 

090 

910 

36 

25 

83  715 

86116 

97  598 

02  402 

35 

25 

84  502 

85  386 

99116 

00  884 

35 

26 

728 

104 

624 

376 

34 

26 

515 

374 

141 

859 

34 

27 

741 

092 

649 

351 

33 

27 

528 

361 

166 

834 

33 

28 

755 

080 

674 

326 

32 

28 

540 

349 

191 

809 

32 

29 

768 

068 

700 

300 

31 

29 

553 

337 

217 

783 

31 

30 

83  781 

86  056 

97  725 

02  275 

30 

30 

84  566 

85  324 

99  242 

00  758 

30 

31 

795 

044 

750 

250 

29 

31 

579 

312 

267 

733 

29 

32 

808 

032 

776 

224 

28 

32 

592 

299 

293 

707 

28 

33 

821 

020 

801 

199 

27 

33 

605 

287 

318 

682 

27 

34 

834 

86  008 

826 

174 

26 

34 

618 

274 

343 

657 

26 

35 

83  848 

85  996 

97  851 

02149 

25 

35 

84  630 

85  262 

99  368 

00  632 

25 

36 

861 

984 

877 

123 

24 

36 

643 

250 

394 

606 

24 

37 

874 

972 

902 

098 

23 

37 

656 

237 

419 

581 

23 

38 

887 

960 

927 

073 

22 

38 

669 

225 

444 

556 

22 

39 

901 

948 

953 

047 

21 

39 

682 

212 

469 

531 

21 

40 

83  914 

85  936 

97978 

02  022 

20 

40 

84  694 

85  200 

99  495 

00  505 

20 

41 

927 

924 

98  003 

01997 

19 

41 

707 

187 

520 

480 

19 

42 

940 

912 

029 

971 

18 

42 

720 

175 

545 

455 

18 

43 

954 

900 

054 

946 

17 

43 

733 

162 

570 

430 

17 

44 

967 

888 

079 

921 

16 

44 

745 

150 

596 

404 

16 

45 

83  980 

85  876 

98  104 

01  896 

15 

45 

84  758 

85  137 

99  621 

00  379 

15 

46 

83  993 

864 

130 

870 

14 

46 

771 

125 

646 

354 

14 

47 

84  006 

851 

155 

845 

13 

47 

784 

112 

672 

328 

13 

48 

020 

839 

180 

820 

12 

48 

796 

100 

697 

303 

12 

49 

033 

827 

206 

794 

11 

49 

809 

087 

722 

278 

11 

50 

84  046 

85  815 

98  231 

01  769 

10 

50 

84  822 

85  074 

99  747 

00  253 

10 

51 

059 

803 

256 

744 

9 

51 

835 

062 

773 

227 

9 

52 

072 

791 

281 

719 

8 

52 

847 

049 

798 

202 

8 

53 

085 

779 

307 

693 

7 

53 

860 

037 

823 

177 

7 

,54 

098 

766 

332 

668 

6 

54 

873 

024 

848 

152 

6 

55 

84112 

85  754 

98  357 

01643 

5 

55 

84  885 

85  012 

99  874 

00126 

5 

56 

125 

742 

383 

617 

4 

56 

898 

84  999 

899 

101 

4 

57 

138 

730 

408 

592 

3 

57 

911 

986 

924 

076 

3 

58 

151 

718 

433 

567 

2 

58 

923 

974 

949 

051 

2 

59 

164 

706 

458 

542 

1 

59 

936 

961 

975 

025 

1 

60 

84177 

9 

85  693 

9 

98  484 

9 

01516 

10 

0 

60 

84  949 

9 

84  949 

9 

00  000 

10 

00  000 

10 

0 

f 

log  cos  ' 

log  sin 

log  cot 

log  tan 

f 

log  cos 

log  sin 

log  cot 

log  tan 

f 

46 


45 


78 


TABLE  YII 


FOR  DETERMINING  THE  FOLLOWING  WITH  GREATER 
ACCURACY  THAN  CAN  BE  DONE  BY  MEANS  OF  TABLE  VI 

1.  log  sin,  log  tan,  and  log  cot,  when  the  angle  is  between  0°  and  2° ; 

2.  log  cos,  log  tan,  and  log  cot,  when  the  angle  is  between  88°  and  90° ; 

3.  The  value  of  the  angle  when  the  logarithm  of  the  function  does  not 

lie  between  the  limits  8.54  684  and  11.  45  316. 


FOKMULAS  FOE  THE  USE  OF  THE  NUMBEES  S AND  T 
I.  When  the  angle  a is  between  0°  and  2° : 


log  sin  a = log  a"  + S. 
log  tan  a = log  a"  + T. 
log  cot  a = colog  tan  a. 


log  ct"  = log  sin  a — S 
= log  tan  a — T 
= colog  cot  a — T. 


II.  When  the  angle  a is  between  88°  and  90° : 


log  cos  a = log  (90°  — a)"  + S. 
log  cot  or  = log  (90°  — a)"  + T. 
log  tan  a — colog  cot  a. 


log  (90°  — a)"  = log  cos  a — S 
= log  cot  a — T 
= colog  tan  a — T; 
O'  90°  - (90°  - a). 


Values  of  S and  T 


0 

2 409 

3 417 

3 823 
4190 

4 840 

5 414 

5 932 

6 408 
6 633 

6 851 

7 267 


4.  68  557 
4.  68  556 
4.  68  555 
4.  68  551 
4.  68  554 
4.  68  553 
4.  68  552 
4.  68  551 
4. 68  550 
4.  68  550 
4.  68  549 

S 


log  sin  a 

8.  06  740 
8.  21  920 
8.  26  795 
8.  30  776 
8.  37  038 
8.  41  904 
8.  45  872 
8.  49  223 
8.  50  721 
8.  52  125 
8.  54  684 
log  sin  a 


0 

200 

1 726 

2 432 

2 976 

3 434 

3 838 

4 204 
4 540 
4 699 

4 853 

5 146 


4.  68  557 
4.  68  558 
4. 68  559 
4. 68  560 
4.  68  561 
4.  68  562 
4.  68  563 
4.  68  564 
4.  68  561 
4.  68  565 
4.  68  566 


log  tan  a 

6.  98  660 

7.  92  263 

8.  07  156 
8. 15  924 
8.  22  142 
8.  26  973 
8.  30  930 
8.  34  270 
8.  35  766 
8.  37  167 
8.  39  713 
log  tan  a 


5 146 
5 424 
5 689 

5 941 
6184 

6 417 
6 642 

6 859 

7 070 
7 173 
7 274 


4.  68  567 
4.  68  568 
4.  68  569 
4.68  570 
4.68  571 
4.  68  572 
4.  68  573 
4.  68  574 
4.  68  571 
4.  68  575 


log  tana 
8. 39  713 
8. 41999 
8.  44  072 
8.  45  955 
8.  47  697 
8.  49  305 
S.  50  802 
8.  52  200 
8.  53  516 
8.  54  145 
8.  54  753 

log  tan  a 


0 


79 


TABLE  YIII 


NATURAL  FUNCTIONS 

Owing  to  the  rapid  change  in  the  functions,  interpolation  is  not 
accurate  for  the  cotangents  from  0°  to  3°,  nor  for  the  tangents  from  87° 
to  90°.  For  the  same  functions  interpolation  is  not  accurate,  in  general, 
in  the  last  figure  from  3°  to  6°  and  from  84°  to  87°,  respectively. 


0°  0° 


f 

sin 

cos 

tan 

cot 

f 

t 

sin 

cos 

tan 

cot 

r 

0 

0.0000 

1.0000 

0.0000 

Infinite 

60 

30 

0.0087 

1.0000 

0.0087 

114.589 

30 

1 

03 

00 

03 

3437.75 

59 

31 

90 

00 

90 

110.892 

29 

2 

06 

00 

06 

1718.87 

58 

32 

93 

00 

93 

107.426 

28 

3 

09 

00 

09 

1145.92 

57 

33 

96 

00 

96 

104.171 

27 

4 

12 

00 

12 

859.436 

56 

34 

99 

1.0000 

99 

101.107 

26 

5 

0.0015 

1.0000 

0.0015 

687.549 

55 

35 

0.0102 

0.9999 

0.0102 

98.2179 

25 

6 

17 

00 

17 

572.957 

54 

36 

05 

99 

05 

95.4895 

24 

7 

20 

00 

20 

491.106 

S3 

37 

08 

99 

08 

92.9085 

23 

8 

23 

00 

23 

429.718 

52 

38 

11 

99 

11 

90.4633 

22 

9 

26 

00 

26 

381.971 

51 

39 

13 

99 

13 

88.1436 

21 

10 

0.0029 

1.0000 

0.0029 

343.774 

50 

40 

0.0116 

0.9999 

0.0116 

85.9398 

20 

11 

32 

00 

32 

312.521 

49 

41 

19 

99 

19 

83.8435 

19 

12 

35 

00 

35 

286.478 

48 

42 

22 

99 

22 

81.8470 

18 

13 

38 

00 

38 

264.441 

47 

43 

25 

99 

25 

79.9434 

17 

14 

41 

00 

41 

245.552 

46 

44 

28 

99 

28 

78.1263 

16 

15 

0.0044 

1.0000 

0.0044 

229.182 

45 

45 

0.0131 

0.9999 

0.0131 

76.3900 

15 

16 

47 

00 

47 

214.858 

44 

46 

34 

99 

34 

74.7292 

14 

17 

49 

00 

49 

202.219 

43 

47 

37 

99 

37 

73.1390 

13 

18 

52 

00 

52 

190.984 

42 

48 

40 

99 

40 

71.6151 

12 

19 

55 

00 

55 

180.932 

41 

49 

43 

99 

43 

70.1533 

11 

20 

0.0058 

1.0000 

0.0058 

171.885 

40 

50 

0.0145 

0.9999 

0.0145 

68.7501 

10 

21 

61 

00 

61 

163.700 

39 

51 

48 

99 

48 

67.4019 

9 

22 

64 

00 

64 

156.259 

38 

52 

51 

99 

51 

66.1055 

8 

23 

67 

00 

67 

149.465 

37 

S3 

54 

99 

54 

64.8580 

7 

24 

70 

00 

70 

143.237 

36 

54 

57 

99 

57 

63.6567 

6 

25 

0.0073 

1.0000 

0.0073 

137.507 

35 

o5 

0.0160 

0.9999 

0.0160 

62.4992 

5 

26 

76 

00 

76 

132.219 

34 

56 

63 

99 

63 

61.3829 

4 

27 

79 

00 

79 

127.321 

33 

57 

66 

99 

66 

60.3058 

3 

28 

81 

00 

81 

122.774 

32 

58 

69 

99 

69 

59.2659 

2 

29 

84 

00 

84 

118.540 

31 

59 

72 

99 

72 

58.2612 

1 

30 

0.0087 

1.0000 

0.0087 

114.589 

30 

60 

0.0175 

0.9998 

0.0175 

57.2900 

O 

/ 

COS 

sin 

cot 

tan 

r 

t 

COS 

sin 

cot 

tan 

f 

89 


89' 


80  1° 


/ 

sin 

cos 

tan 

cot 

/ 

o 

0.0175 

0.9998 

0.0175 

57.2900 

60 

1 

77 

98 

77 

56.3506 

59 

2 

80 

98 

80 

55.4415 

58 

3 

83 

98 

83 

54.5613 

57 

4 

86 

98 

86 

53.7086 

56 

5 

0.0189 

0.9998 

0.0189 

52.8821 

55 

6 

92 

98 

92 

52.0807 

54 

7 

95 

98 

95 

51.3032 

53 

8 

0198 

98 

0198 

50.5485 

52 

9 

0201 

98 

0201 

49.8157 

51 

10 

0.0204 

0.9998 

0.0204 

49.1039 

50 

11 

07 

98 

07 

48.4121 

49 

12 

09 

98 

09 

47.7395 

48 

13 

12 

98 

12 

47.0853 

47 

14 

15 

98 

15 

46.4489 

46 

15 

0.0218 

0.9998 

0.0218 

45.8294 

45 

16 

21 

98 

21 

45.2261 

44 

17 

24 

97 

24 

44.6386 

43 

18 

27 

97 

27 

44.0661 

42 

19 

30 

97 

30 

43.5081 

41 

20 

0.0233 

0.9997 

0.0233 

42.9641 

40 

21 

36 

97 

36 

42.4335 

39 

22 

39 

97 

39 

41.9158 

38 

23 

41 

97 

41 

41.4106 

37 

24 

44 

97 

44 

40.9174 

36 

25 

0.0247 

0.9997 

0.0247 

40.4358 

35 

26 

50 

97 

50 

39.9655 

34 

27 

53 

97 

53 

39.5059 

33 

28 

56 

97 

56 

39.0568 

32 

29 

59 

97 

59 

38.6177 

31 

30 

0.0262 

0.9997 

0.0262 

38.1885 

30 

31 

65 

96 

65 

37.7686 

29 

32 

68 

96 

68 

37.3579 

28 

33 

70 

96 

71 

36.9560 

27 

34 

73 

96 

74 

36.5627 

26 

35 

0.0276 

0.9996 

0.0276 

36.1776 

25 

36 

79 

96 

79 

35.8006 

24 

37 

82 

96 

82 

35.4313 

23 

38 

85 

96 

85 

35.0695 

22 

39 

88 

96 

88 

34.7151 

21 

40 

0.0291 

0.9996 

0.0291 

34.3678 

20 

41 

94 

96 

94 

34.0273 

19 

42 

0297 

96 

0297 

33.6935 

18 

43 

0300 

96 

0300 

33.3662 

17 

44 

02 

95 

03 

33.0452 

16 

45 

0.0305 

0.9995 

0.0306 

32.7303 

15 

46 

08 

95 

08 

32.4213 

14 

47 

11 

95 

11 

32.1181 

13 

48 

14 

95 

14 

31.8205 

12 

49 

17 

95 

17 

31.5284 

11 

50 

0.0320 

0.9995 

0.0320 

31.2416 

10 

51 

23 

95 

23 

30.9599 

9 

52 

26 

95 

26 

30.6833 

8 

53 

29 

95 

29 

30.4116 

7 

54 

32 

95 

32 

30.1446 

6 

55 

0.0334 

0.9994 

0.0335 

29.8823 

5 

56 

37 

94 

38 

29.6245 

4 

57 

40 

94 

40 

29.3711 

3 

58 

43 

94 

43 

29.1220 

2 

59 

46 

94 

46 

28.8771 

1 

60 

0.0349 

0.9994 

0.0349 

28.6363 

0 

f 

COS 

sin 

cot 

tan 

/ 

2“ 


t 

sin 

cos 

tan 

cot 

/ 

o 

0.0349  0.9994 

0.0349 

28.6363 

60 

1 

52 

94 

52 

28.3994 

59 

2 

55 

94 

55 

28.1664 

58 

3 

58 

94 

58 

27.9372 

57 

4 

61 

93 

61 

27.7117 

56 

5 

0.0364 

0.9993 

0.0364 

27.4899 

55 

6 

66 

93 

67 

27.2715 

54 

7 

69 

93 

70 

27.0566 

53 

8 

72 

93 

73 

26.8450 

52 

9 

75 

93 

75 

26.6367 

51 

lO 

0.0378 

0.9993 

0.0378 

26.4316 

oO 

11 

81 

93 

81 

26.2296 

49 

12 

84 

93 

84 

26.0307 

48 

13 

87 

93 

87 

25.8348 

47 

14 

90 

92 

90 

25.6418 

46 

15 

0.0393 

0.9992 

0.0393 

25.4517 

45 

16 

96 

92 

96 

25.2644 

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41 

16 

632 

38 

23 

09 

39 

21 

624 

37 

24 

11 

37 

26 

617 

36 

25 

0.6214 

0.7835 

0.7931  1.2609 

35 

26 

16 

33 

35 

602 

34 

27 

18 

32 

40 

594 

33 

28 

21 

30 

45 

587 

32 

29 

23 

28 

50 

579 

31 

30 

0.6225 

0.7826 

0.7954  1.2572 

30 

31 

27 

24 

59 

564 

29 

32 

30 

22 

64 

557 

28 

33 

32 

21 

69 

549 

27 

34 

34 

19 

73 

542 

26 

35 

0.6237 

0.7817 

0.7978  1.2534 

25 

36 

39 

IS 

83 

527 

24 

37 

41 

13 

88 

519 

23 

38 

43 

12 

92 

512 

22 

39 

46 

10 

7997 

504 

21 

40 

0.6248 

0.7808 

0.8002  1.2497 

20 

41 

50 

06 

07 

489 

19 

42 

52 

04 

12 

482 

18 

43 

55 

02 

16 

475 

17 

44 

57 

7801 

. 21 

467 

16 

45 

0.6259 

0.7799 

0.8026  1.2460 

15 

46 

62 

97 

31 

452 

14 

47 

64 

95 

35 

445 

13 

48 

66 

93 

40 

437 

12 

49 

68 

92 

45 

430 

11 

50 

0.6271 

0.7790 

0.8050  1.2423 

10 

51 

73 

88 

55 

415 

9 

52 

75 

86 

59 

408 

8 

53 

77 

84 

64 

401 

7 

54 

80 

82 

69 

393 

6 

55 

0.6282 

0.7781 

0.8074  1.2386 

5 

56 

84 

79 

79 

378 

4 

57 

86 

77 

83 

371 

3 

58 

89 

75 

88 

364 

2 

59 

91 

73 

93 

356 

1 

60 

0.6293 

0.7771 

0.8098  1.2349 

o 

f 

COS 

sin 

cot 

tan 

t 

52 


51 


39® 


40°  99 


/ 

sin 

cos 

tan 

cot 

t 

o 

0.6428 

0.7660 

0.8391 

1.1918 

60 

1 

30 

59 

8396 

910 

59 

2 

32 

57 

8401 

903 

58 

3 

35 

55 

06 

896 

57 

4 

37 

53 

11 

889 

56 

5 

0.6439 

0.7651 

0.8416 

1.1882 

55 

6 

41 

49 

21 

875 

54 

7 

43 

47 

26 

868 

53 

8 

46 

45 

31 

861 

52 

9 

48 

44 

36 

854 

51 

lO 

0.6450 

0.7642 

0.8441 

1.1847 

50 

11 

52 

40 

46 

840 

49 

12 

55 

38 

51 

833 

48 

13 

57 

36 

56 

826 

47 

14 

59 

34 

61 

819 

46 

15 

0.6461 

0.7632 

0.8466 

1.1812 

45 

16 

63' 

30 

71 

806 

44 

17 

66 

29 

76 

799 

43 

18 

68 

27 

81 

792 

42 

19 

70 

25 

86 

785 

41 

20 

0.6472 

0.7623 

0.8491 

1.1778 

40 

21 

75 

21 

8496 

771 

39 

22 

77 

19 

8501 

764 

38 

23 

79 

17 

06 

757 

37 

24 

81 

15 

11 

750 

36 

25 

0.6483 

0.7613 

0.8516 

1.1743 

35 

26 

86 

12 

21 

736 

34 

27 

88 

10 

26 

729 

33 

28 

90 

08 

31 

722 

32 

29 

92 

06 

36 

715 

31 

30 

0.6494 

0.7604 

0.8541 

1.1708 

30 

31 

97 

02 

46 

702 

29 

32 

6499 

7600 

51 

695 

28 

33 

6501 

7598 

56 

688 

27 

34 

03 

96 

61 

681 

26 

35 

0.6506 

0.7595 

0.8566 

1.1674 

25 

36 

08 

93 

71 

667 

24 

37 

10 

91 

76 

660 

23 

38 

12 

89 

81 

653 

22 

39 

14 

87 

86 

647 

21 

40 

0.6517 

0.7585 

0.8591 

1.1640 

20 

41 

19 

83 

8596 

633 

19 

42 

21 

81 

8601 

626 

18 

43 

23 

79 

06 

619 

17 

44 

25 

78 

11 

612 

16 

45 

0.6528 

0.7576 

0.8617 

1.1606 

15 

46 

30 

74 

22 

599 

14 

47 

32 

72 

27 

592 

13 

48 

34 

70 

32 

585 

12 

49 

36 

68 

37 

578 

11 

50 

0.6539 

0.7566 

0.8642 

1.1571 

10 

51 

41 

64 

47 

565 

9 

52 

43 

62 

52 

558 

8 

53 

45 

60 

57 

551 

7 

54 

47 

59 

62 

544 

6 

55 

0.6550 

0.7557 

0.8667 

1.1538 

5 

56 

52 

55 

72 

531 

4 

57 

54 

53 

78 

524 

3 

58 

56 

51 

83 

517 

2 

59 

58 

49 

88 

510 

1 

60 

0.6561 

0.7547 

0.8693 

1.1504 

0 

f 

COS 

sin 

cot 

tan 

50' 


49 


100  41® 


42® 


f 

sin 

cos 

tan 

cot 

f 

0 

0.6691 

0.7431 

0.9004 

1.1106 

60 

1 

93 

30 

09 

100 

59 

2 

96 

28 

15 

093 

58 

3 

6698 

26 

20 

087 

57 

4 

6700 

24 

25 

080 

56 

5 

0.6702 

0.7422 

0.9030 

1.1074 

55 

6 

04 

20 

36 

067 

54 

7 

06 

18 

41 

061 

53 

8 

09 

16 

46 

054 

52 

9 

11 

14 

52 

048 

51 

10 

0.6713 

0.7412 

0.9057 

1.1041 

50 

11 

15 

10 

62 

035 

49 

12 

17 

08 

67 

028 

48 

13 

19 

06 

73 

022 

47 

14 

22 

04 

78 

016 

46 

15 

0.6724 

0.7402 

0.9083 

1.1009 

45 

16 

26 

7400 

89 

1.1003 

44 

17 

28 

7398 

94 

1.0996 

43 

18 

30 

96 

9099 

990 

42 

19 

32 

94 

9105 

983 

41 

20 

0.6734 

0.7392 

0.9110 

1.0977 

40 

21 

37 

90 

15 

971 

39 

22 

39 

88 

21 

964 

38 

23 

41 

87 

26 

958 

37 

24 

43 

85 

31 

951 

36 

25 

0.6745 

0.7383 

0.9137 

1.0945 

35 

26 

47 

81 

42 

939 

34 

27 

49 

79 

47 

932 

33 

28 

52 

77 

53 

926 

32 

29 

54 

75 

58 

919 

31 

30 

0.6756 

0.7373 

0.9163 

1.0913 

30 

31 

58 

71 

69 

907 

29 

32 

60 

69 

74 

900 

28 

33 

62 

67 

79 

894 

27 

34 

64 

65 

85 

888 

26 

35 

0.6767 

0.7363, 

0.9190 

1.0881 

25 

36 

69 

61 

9195 

875 

24 

37 

71 

59 

9201 

869 

23 

38 

73 

57 

06 

862 

22 

39 

75 

55 

12 

856 

21 

40 

0.6777 

0.7353 

0.9217 

1.0850 

20 

41 

79 

51 

22 

843 

19 

42 

82 

49 

28 

837 

IS 

43 

84 

47 

33 

831 

17 

44 

86 

45 

• 39 

824 

16 

45 

0.6788 

0.7343 

0.9244 

1.0818 

15 

46 

90 

41 

49 

812 

14 

47 

92 

39 

55 

805 

13 

48 

94 

37 

60 

799 

12 

49 

97 

35 

66 

793 

11 

50 

0.6799 

0.7333 

0.9271 

1.0786 

10 

51 

6801 

31 

76 

780 

9 

52 

03 

29 

82 

774 

8 

53 

05 

27 

87 

768 

7 

54 

07 

25 

93 

761 

6 

5o 

0.6809 

0.7323 

0.9298 

1.0755 

5 

56 

11 

21 

9303 

749 

4 

57 

14 

19 

09 

742 

3 

58 

16 

18 

14 

736 

2 

59 

18 

16 

20 

730 

1 

60 

0.6820 

0.7314 

0.9325 

1.0724 

0 

/ 

COS 

sin 

cot 

tan 

f 

48 


47 


43° 


t 

sin 

cos 

tan 

cot 

/ 

o 

0.6820 

0.7314 

0.9325 

1.0724 

60 

1 

22 

12 

31 

717 

59 

2 

24 

10 

36 

711 

58 

3 

26 

08 

41 

705 

57 

4 

28 

06 

47 

699 

56 

5 

0.6831 

0.7304 

0.9352 

1.0692 

55 

6 

33 

02 

58 

686 

54 

7 

35 

7300 

63 

680 

53 

8 

37 

7298 

69 

674 

52 

9 

39 

96 

74 

668 

51 

lO 

0.6841 

0.7294 

0.9380 

1.0661 

50 

11 

43 

92 

85 

655 

49 

12 

45 

90 

91 

649 

48 

13 

48 

88 

9396 

643 

47 

14 

50 

86 

9402 

637 

46 

15 

0.6852 

0.7284 

0.9407 

1.0630 

45 

16 

54 

82 

13 

624 

44 

17 

56 

80 

18 

618 

43 

18 

58 

78 

24 

612 

42 

19 

60 

76 

29 

606 

41 

20 

0.6862 

0.7274 

0.9435 

1.0599 

40 

21 

65 

72 

40 

593 

39 

22 

67 

70 

46 

587 

38 

23 

69 

68 

51 

581 

37 

24 

71 

66 

57 

575 

36 

25 

0.6873 

0.7264 

0.9462 

1.0569 

35 

26 

75 

62 

68 

562 

34 

27 

77 

60 

73 

556 

33 

28 

79 

58 

79 

550 

32 

29 

81 

56 

84 

544 

31 

30 

0.6884 

0.7254 

0.9490 

1.0538 

30 

31 

86 

52 

9495 

532 

29 

32 

88 

50 

9501 

526 

28 

33 

90 

48 

06 

519 

27 

34 

92 

46 

12 

513 

26 

35 

0.6894 

0.7241 

0.9517 

1.0507 

25 

36 

96 

42 

23 

501 

24 

37 

6898 

40 

28 

495 

23 

38 

6900 

38 

34 

489 

22 

39 

03 

36 

40 

483 

21 

40 

0.6905 

0.7234 

0.9545 

1.0477 

20 

41 

07 

32 

51 

470 

19 

42 

09 

30 

56 

464 

18 

43 

11 

28 

62 

458 

17 

44 

13 

26 

67 

452 

16 

45 

0.6915 

0.7224 

0.9573 

1.0446 

15 

46 

17 

22 

78 

440 

14 

47 

19 

20 

84 

434 

13 

48 

21 

18 

90 

428 

12 

49 

24 

16 

9595 

422 

11 

50 

0.6926 

0.7214 

0.9601 

1.0416 

10 

51 

28 

12 

06 

410 

9 

52 

30 

10 

12 

404 

8 

53 

32 

08 

18 

398 

7 

54 

34 

06 

23 

392 

6 

55 

0.6936 

0.7203 

0.9629 

1.0385 

5 

56 

38 

7201 

34 

379 

4 

57 

40 

7199 

40 

373 

3 

58 

42 

97 

46 

367 

2 

59 

44 

95 

51 

361 

1 

60 

0.6947 

0.7193 

0.9657 

1.0355 

0 

COS 

sin 

cot 

tan 

/ 

44°  101 


sin 

cos 

tan 

cot 

/ 

o 

0.6947 

0.7193 

0.9657 

1.0355 

60 

1 

49 

91 

63 

349 

59 

2 

51 

89 

68 

343 

58 

3 

53 

87 

74 

337 

57 

4 

55 

85 

79 

331 

56 

5 

0.6957 

0.7183 

0.9685 

1.0325 

55 

6 

59 

81 

91 

319 

54 

7 

61 

79 

9696 

313 

53 

8 

63 

77 

9702 

307 

52 

9 

65 

75 

08 

301 

51 

10 

0.6967 

0.7173 

0.9713 

1.0295 

50 

11 

70 

71 

19 

289 

49 

12 

72 

69 

25 

283 

48 

13 

74 

67 

30 

277 

47 

14 

76 

65 

36- 

271 

46 

15 

0.6978 

0.7163 

0.9742 

1.0265 

45 

16 

80 

61 

47 

259 

44 

17 

82 

59 

53 

253 

43 

18 

84 

57 

59 

247 

42 

19 

86 

55 

64 

241 

41 

20 

0.6988 

0.7153 

0.9770 

1.0235 

40 

21 

90 

51 

,76 

230 

39 

22 

92 

49 

81 

224 

38 

23 

95 

47 

87 

218 

37 

24 

97 

45 

93 

212 

36 

25 

0.6999 

0.7143 

0.9798 

1.0206 

35 

26 

7001 

41 

9804 

200 

34 

27 

03 

39 

10 

194 

33 

28 

05 

37 

16 

188 

32 

29 

07 

35 

21 

182 

31 

30 

0.7009 

0.7133 

0.9827 

1.0176 

30 

31 

11 

30 

33 

170 

29 

32 

13 

28 

38 

164 

28 

33 

15 

26 

44 

158 

27 

'34 

17 

24 

50 

152 

26 

35 

0.7019 

0.7122 

0.9856 

1.0147 

25 

36 

22 

20 

61 

141 

24 

37 

24 

18 

67 

135 

23 

38 

26 

16 

73 

129 

22 

39 

28 

14 

79 

123 

21 

40 

0.7030 

0.7112 

0.9884 

1.0117 

20 

41 

32 

10 

90 

111 

19 

42 

34 

08 

9896 

105 

18 

43 

36 

06 

9902 

099 

17 

44 

38 

04 

07 

094 

16 

45 

0.7040 

0.7102 

0.9913 

1.0088 

15 

46 

42 

7100 

19 

082 

14 

47 

44 

7098 

25 

076 

13 

48 

46 

96 

30 

070 

12 

49 

48 

94 

36 

064 

11 

50 

0.7050 

0.7092 

0.9942 

1.0058 

10 

51 

53 

90 

48 

052 

9 

52 

55 

88 

54 

047 

8 

53 

57 

85 

59 

041 

7 

54 

59 

83 

65 

035 

6 

55 

0.7061 

0.7081 

0.9971 

1.0029 

5 

56 

63 

79 

77 

023 

4 

57 

65 

77 

83 

017 

3 

58 

67 

75 

88 

012 

2 

59 

69 

73 

94 

006 

1 

60 

0.7071 

0.7071 

1.0000 

1.0000 

0 

t 

COS 

sin 

cot 

tan 

/ 

46 


45° 


102 


TABLE  IX 


CONVERSION  TABLE— DEGREES  TO  RADIANS 


1°  = radians  1 radian  = — degrees 

180  7t  ^ 


0°-45° 


o 

O' 

lO' 

20' 

30' 

40' 

50' 

o 

0.0000 

0.0029 

0.0058 

0.0087 

0.0116 

0.0145 

1 

0175 

0204 

0233 

0262 

0291 

0320 

2 

0349 

0378 

0407 

0436 

0165 

0495 

3 

0524 

0553 

0582 

0611 

0640 

0669 

4 

0698 

0727 

0756 

0785 

0814 

0844 

5 

0.0873 

0.0902 

0.0931 

0.0960 

0.0989 

0.1018 

6 

1047 

1076 

1105 

1134 

1164 

1193 

7 

1222 

1251 

1280 

1309 

1338 

1367 

8 

1396 

1425 

1454 

1484 

1513 

1542 

9 

1571 

1600 

1629 

1658 

1687 

1716 

lO 

0.1745 

0.1774 

0.1804 

0.1833 

0.1862 

0.1891 

11 

1920 

1949 

1978 

2007 

2036 

2065 

12 

2094 

2123 

2153 

2182 

2211 

2240 

13 

2269 

2298 

2327 

2356 

2385 

2414 

14 

2443 

2473 

2502 

2531 

2560 

2589 

15 

0.2618 

0.2647 

0.2676 

0.2705 

0.2734 

0.2763 

16 

2793 

2822 

2851 

2880 

2909 

2938 

17 

2967 

2996 

3025 

3054 

3083 

3113 

18 

3142 

3171 

3200 

3229 

3258 

3287 

19 

3316 

3345 

3374 

3403 

3432 

3462 

20 

0.3491 

0.3520 

0.3549 

0.3578 

0.3607 

0.3636 

21 

3665 

3694 

3723 

3752 

3782 

3811 

22 

3840 

3869 

3898 

3927 

3956 

3985 

23 

4014 

4043 

4072 

4102 

4131 

4160 

24 

4189 

4218 

4247 

4276 

4305 

4334 

25 

0.4363 

0.4392 

0.4422 

0.4451 

0.4480 

0.4508 

26 

4538 

4567 

4596 

4625 

4654 

4683 

27 

4712 

4741 

4771 

4800 

4829 

4858 

28 

4887 

4916 

4945 

4974 

5003 

5032 

29 

5061 

5091 

5120 

5149 

5178 

5207 

50 

0.5236 

0.5265 

0.5294 

0.5323 

0.5352 

0.5381 

31 

5411 

5440 

5469 

5498 

5527 

5556 

32 

5585 

5614 

5643 

5672 

5701 

5730 

33 

5760 

5789 

5818 

5847 

5876 

5905 

34 

5934 

5963 

5992 

6021 

6050 

6080 

35 

0.6109 

0.6138 

0.6167 

0.6196 

0.6225 

0.6254 

36 

6283 

6312 

6341 

6370 

6400 

6429 

37 

6458 

6487 

6516 

6545 

6574 

6603 

38 

6632 

6661 

6690 

6720 

6749 

6778 

39 

6807 

6836 

6865 

6894 

6923 

6952 

40 

0.6981 

0.7010 

0.7039 

0.7069 

0.7098 

0.7127 

41 

7156 

7185 

7214 

7243 

7272 

7301 

42 

7330 

■ 7359 

7389 

7418 

7447 

7476 

43 

7505 

7534 

7563 

7592 

7621 

7650 

44 

7679 

7709 

7738 

7767 

7796 

7825 

45 

0.7854 

0.7883 

0.7912 

0.7941 

0.7970 

0.7999 

O 

O' 

10' 

20' 

30' 

40' 

50' 

103 


In  using  this  table,  interpolations  may  be  made  as  with  other  tables. 
Thus  to  find  the  number  of  radians  corresponding  to  49°  15',  we  have : 

49°  10'  = 0.8581  radians 
Tabular  diff.  = 0.0029 
A of  0.0029  = 0.0015 
Adding,  49°  15'  = 0.8596  radians 

45°-90° 

o 

O' 

10' 

20' 

30' 

40' 

50' 

45 

0.7854 

0.7883 

0.7912 

0.7941 

0.7970 

0.7999 

46 

8029 

8058 

8087 

. 8116 

8145 

8i74 

47 

8203 

8232 

8261 

8290 

8319 

8348 

48 

8378 

8407 

8436 

8465 

8494 

8523 

49 

8552 

8581 

8610 

8639 

8668 

8698 

50 

0.8727 

0.8756 

0.8785 

0.8814 

0.8843 

0.8872 

51 

8901 

8930 

8959 

8988 

9018 

9047 

52 

9076 

9105 

9134 

9163 

9192 

9221 

53 

9250 

9279 

9308 

9338 

9367 

9396 

54 

9425 

9454 

9483 

9512 

9541 

9570 

55 

0.9599 

0.9628 

0.9657 

0.9687 

0.9716 

0.9745 

56 

9774 

9803 

9832 

9861 

9890 

9919 

57 

9948 

9977 

1.0007 

1.0036 

1.0065 

1.0094 

58 

1.0123 

1.0152 

0181 

0210 

0239 

0268 

59 

0297 

0326 

0356 

0385 

0414 

0443 

60 

1.0472 

1.0501 

1.0530 

1.0559 

1.0588 

1.0617 

61 

0647 

0676 

0705 

0734 

0763 

0792 

62 

0821 

0850 

0879 

0908 

0937 

0966 

63 

0996 

1025 

1054 

1083 

1112 

1141 

64 

1170 

1199 

1228 

1257 

1286 

1316 

65 

1.1345 

1.1374 

1.1403 

1.1432 

1.1461 

1.1490 

66 

1519 

1548 

1577 

1606 

1636 

1665 

67 

1694 

1723 

1752 

1781 

1810 

1839 

68 

1868 

1897 

1926 

1956 

1985 

2014 

69 

2043 

2072 

2101 

2130 

2159 

2188 

70 

1.2217 

1.2246 

1.2275 

1.2305 

1.2334 

1.2363 

71 

2392 

2421 

2450 

2479 

2508 

2537 

72 

2566 

2595 

2625 

2654 

2683 

2712 

73 

2741 

2770 

2799 

2828 

2857 

2886 

74 

2915 

2945 

2974 

3003 

3032 

3061 

75 

1.3090 

1.3119 

1.3148 

1.3177 

1.3206 

1.3235 

76 

3265 

3294 

3323 

3352 

3381 

3410 

77 

3439 

3468 

3497 

3526 

3555 

3584 

78 

3614 

3643 

3672 

3701 

3730 

3759 

79 

3788 

3817 

3846 

3875 

3904 

3934 

80 

1.3963 

1.3992 

1.4021 

1.4050 

1.4079 

1.4108 

81 

4137 

4166 

4195 

4224 

4254 

4283 

82 

4312 

4341 

4370 

4399 

4428 

4457 

83 

4486 

4515 

4544 

4573 

4603 

4632 

84 

4661 

4690 

4719 

4748 

4777 

4806 

85 

1.4835 

1.4864 

1.4893 

1.4923 

1.4952 

1.4981 

86 

5010 

5039 

5068 

5097 

5126 

5155 

87 

5184 

5213 

5243 

5272 

5301 

5330 

88 

5359 

5388 

5417 

5446 

5475 

5504 

89 

5533 

5563 

5592 

5621 

5650 

5679 

90 

1.5708 

1.5737 

1.5766 

1.5795 

1.5824 

1.5853 

0 

O' 

lO' 

20' 

30' 

40' 

60' 

104 


TABLE  X.  CONVERSION  OF  MINUTES  AND  SECONDS  TO 

DECIMALS  OF  A DEGREE,  AND  OF  DECIMALS  OF  A DEGREE 

TO  MINUTES  AND 

SECONDS 

t 

0 

n 

O 

0 

> and  " 

o 

1 and  " 

O 

0.0000 

o 

0.00000 

0.000 

0'  0" 

0.50 

30'  0 " 

1 

0167 

1 

028 

001 

0'  4" 

51 

30'  36" 

2 

0333 

2 

056 

002 

0'  7" 

52 

31'  12" 

3 

0500 

3 

083 

003 

0'  11" 

53 

31'  48" 

4 

0667 

4 

111 

004 

0'  14" 

54 

32'  24" 

5 

0.0833 

5 

0.00139 

0.005 

0'  18" 

0.55 

33'  0" 

6 

1000 

6 

167 

006 

O'  22" 

56 

33'  36" 

7 

1167 

7 

194 

007 

0'  25" 

57 

34'  12" 

8 

1333 

8 

222 

008 

0'  29" 

58 

34'  48" 

9 

1500 

9 

250 

009 

0'  32" 

59 

35'  24" 

lO 

0.1667 

lO 

0.00278 

0.00 

0'  0" 

0.60 

36'  0" 

11 

1833 

11 

306 

01 

0'  36" 

61 

36'  36" 

12 

2000 

12 

333 

02 

1'  12" 

62 

37'  12" 

13 

2167 

13 

361 

03 

1'  48" 

63 

37'  48" 

14 

2333 

14 

389 

04 

2'  24" 

64 

38'  24" 

15 

0.2500 

15 

0.00417 

0.05 

3'  0" 

0.65 

39'  0" 

16 

2667 

16 

444 

06 

3'  36" 

66 

39'  36" 

17 

2833 

17 

472 

07 

4'  12" 

67 

40'  12" 

18 

3000 

18 

500 

08 

4'  48" 

68 

40'  48" 

19 

3167 

19 

528 

09 

5'  24" 

69 

41'  24" 

20 

0.3333 

20 

0.00556 

0.10 

6'  0" 

0.70 

42'  0" 

21 

3500 

21 

583 

11 

6'  36" 

71 

42'  36" 

22 

3667 

22 

611 

12 

7'  12" 

72 

43'  12" 

23 

3833 

23 

639 

13 

7'  48" 

73 

43'  48" 

24 

4000 

24 

667 

14 

8'  24" 

74 

44'  24" 

25 

0.4167 

25 

0.00694 

0.15 

9'  0" 

0.75 

45'  0" 

26 

4333 

26 

722 

16 

9'  36" 

76 

45'  36" 

27 

4500 

27 

750 

17 

10'  12" 

77 

46'  12" 

28 

4667 

28 

778 

18 

10'  48" 

78 

46'  48" 

29 

4833 

29 

806 

19 

11'  24" 

79 

47'  24" 

30 

0.5000 

30 

0.00833 

0.20 

12'  0" 

0.80 

48'  0" 

31 

5167 

31 

861 

21 

12'  36" 

81 

48'  36" 

32 

5333 

32 

889 

22 

13'  12" 

82 

49'  12" 

33 

5500 

33 

917 

23 

13'  48" 

83 

49'  48" 

34 

5667 

34 

944 

24 

14'  24" 

84 

50'  24" 

35 

0.5833 

35 

0.00972 

0.25 

15'  0" 

0.85 

51'  0" 

36 

6000 

36 

01000 

26 

15'  36" 

86 

51'  36" 

37 

6167 

37 

028 

27 

16'  12" 

87 

52'  12" 

38 

6333 

38 

056 

28 

16'  48" 

88 

52'  48" 

39 

6500 

39 

083 

29 

17'  24" 

89 

53'  24" 

40 

0.6667 

40 

0.01111 

0.30 

IS'  0" 

0.90 

54'  0" 

41 

6833 

41 

139 

31 

18'  36" 

91 

54'  36" 

42 

7000 

42 

167 

32 

19'  12" 

92 

55'  12" 

43 

7167 

43 

194 

33 

19'  48" 

93 

55'  48" 

44 

7333 

44 

222 

34 

20'  24" 

94 

56'  24" 

45 

0.7500 

45 

0.01250 

0.35 

21'  0" 

0.95 

57'  0" 

46 

7667 

46 

278 

36 

21'  36" 

96 

57'  36" 

47 

7833 

47 

306 

37 

22'  12" 

97 

58'  12" 

48 

8000 

48 

333 

38 

22'  48" 

98 

58'  48" 

49 

8167 

49 

361 

39 

23'  24" 

99 

59'  24" 

50 

0.8333 

50 

0.01389 

0.40 

24'  0" 

1.00 

60'  0" 

51 

8500 

51 

417 

41 

24'  36" 

10 

66'  0" 

52 

8667 

52 

444 

42 

25'  12" 

20 

72'  0" 

53 

8833 

53 

472 

43 

25'  48" 

30 

78'  0" 

54 

9000 

54 

500 

44 

26'  24" 

40 

84'  0" 

55 

0.9167 

55 

0.01528 

0.45 

27'  0" 

1.50 

90'  0" 

56 

9333 

56 

556 

46 

27'  36" 

60 

96'  0" 

. 57 

9500 

57 

583 

47 

28'  12" 

70 

102'  0" 

58 

9667 

58 

611 

48 

28'  48" 

SO 

108'  0" 

59 

9833 

59 

639 

49 

29'  24" 

90 

114'  0" 

60 

1.0000 

60 

0.01667 

0.50 

30'  0" 

2.00 

120'  0" 

f 

0 

tf 

0 

0 

r and  " 

0 

1 and  " 

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